On IP=PSPACE and theorems with narrow proofs

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21世纪大学英语读写教程第一册第3课内容讲解

21世纪大学英语读写教程第一册第3课内容讲解21世纪大学英语读写教程第一册第3课内容讲解导语：史蒂夫莫里斯一出生就是盲目的，可是他并没有自我放弃，而是顽强生存得有声于色，下面是一篇讲述史蒂夫莫里斯的英语课文，欢迎阅读。

Stevie Wonder: Sunshine in the ShadowWhen Stevie Morris was born, on May 13, 1950, the doctors shook their heads and told the mother that her son was born blind and likely would always be that way. She broke into tears.Blind and black and poor — what kind of life could this new infant have? In her wildest dreams, Mrs. Morris could never have imagined that her new baby would become a famous musician called Stevie Wonder. At the time, all she could do was pray —and worry.Stevie himself didn't worry at all. Life was too full. He was brought up among church-going people whose faith helped them bear the poverty. He loved music and would pound spoons or forks on any surface that faintly resembled a drum.He even ran and played with sighted children. "I didn't realize I was blind until I was about four," he says. That might sound strange. To a small child just learning about the world, it wasn't strange at all. Stevie heard and smelled and touched. As far as he knew, that was all anyone could do. That was life.When Stevie's mother got tired of her tables being used for drums, she bought him a toy set. He played so hard that he had actually worn the toy out within a few weeks. Other toy sets followed; then an uncle added a toy harmonica, and Stevie learned to play it so quickly that everyone was amazed.Stevie taught himself to play the piano as quickly as he hadonce learned the harmonica. With friends, he began playing rock and roll music. They performed on the front porch of Stevie's apartment building, drawing crowds of neighbors to watch and listen and clap time to the beat."I loved that beat," Stevie says. He not only loved the beat, he was very good at making it.Ronnie White, of the Miracles singing group, heard Stevie and promptly took him down to his recording company, Motown Records."Give him an audition," Ronnie said. They did. All the top people at Motown got together to hear a little blind boy who wasn't even ten years old yet. At first, they were being nice. Poor kid. They didn't want to hurt his feelings.Then they heard Stevie sing and play, and nobody said "poor kid" anymore. They were too busy congratulating themselves on finding a youngster who could be the musical talent of the decade. "He's a wonder boy," somebody said as they watched little Stevie dart from one instrument to the next, playing each one with ease."Wonder," somebody else said, "Little Stevie Wonder."The new name stuck and Stevie Morris became Little Stevie Wonder. He had his first hit when he was twelve years old. It was called "Fingertips" and it was a smash.Over the following years, Little Stevie Wonder became one of the top recording artists at Motown, producing one hit after another. But as he grew into adulthood, Stevie began to get tired of the way the Motown company controlled all aspects of his career. He wanted to write and produce his own songs, but the Motown company thought it was unwise to change a winning formula.When he turned 21, Stevie finally got his freedom. Against Motown's wishes he started exploring: he made records that combined gospel, rock and roll, and jazz and which used African and Latin American rhythms. To the record company's surprise, Stevie's new albums such as "Music of My Mind" and "Innervisions" were even more popular than his early ones. Stevie Wonder had become a mature man and an independent musical artist.Just after this success, however, tragedy struck. In August of 1973, Stevie was involved in a serious car accident. For nearly a week he lay in a coma, unable to speak or walk. "We don't know when he'll be out of danger," the doctor said. Everyone waited and prayed. Suddenly, it didn't matter that Stevie was a musical genius or that he had conquered blindness and poverty. All he had left was his faith and strong will.That turned out to be enough. Stevie fought back from the shadow of death as he had once fought out from the shadow of blindness. He went on to give more performances, make more hit records.The car accident changed Stevie by making him reevaluate his goals in life. He still loved to make music, but he also started to pay more attention to the world outside. He worked to create a national holiday to honor the civil rights leader Dr.Martin Luther King, Jr.. He recorded songs urging racial harmony and raised money to end world hunger. Recently, Stevie was honored by South African president Nelson Mandela for his work against that country's system of racial apartheid.Stevie Wonder has faith and fame, wealth and love. He has not only conquered his own darkness, but through his music and his social activities he has been able to bring sunshine to theshadow of many other lives.New Wordssunshinen. the light and heat of the sun 阳光infantn. a very young child 婴儿musiciann. a person who performs on a musical instrument, or who writes music 乐师，作曲家prayvi. (for, to) speak to God in order to give thanks or to ask for help 祈祷，祈求spoonn. 匙，调羹faintlhad. slightly; mildly 轻微地;微弱地resemblevt. look or be like 像，类似drumn. 鼓tirev. (使)感到疲劳;(使)厌倦，(使)厌烦harmonican. 口琴amazevt. fill with great surprise; cause wonder in 使惊奇;使惊羡*porchn. (建筑物前有顶的)门廊，入口处apartmentn. 公寓大楼;一套公寓房间vi. applaud 拍手miraclen. 奇迹promptlyad. immediately and without any delay 迅速地，及时地auditionn. (对志愿艺人等的.)面试(指试读、试唱、试奏等)kidn. a child 小孩congratulatevt. speak to (a person) with praise and admiration for a happy event or sth. successfully done 祝贺youngstern. a young person, esp. a boy 年轻人;男孩talentn. 1. a special ability or skill 天才，天资;超常智能2. people of such ability 人才decaden. a period of 10 years 十年(期)*dartvi. move suddenly and quickly 猛冲，飞奔instrumentn. 乐器;仪器;器具;器械easen. 1. freedom from difficulty 容易2. freedom from discomfort, pain or worry 安适;悠闲;无痛苦;无忧虑fingertipn. the end of a finger 指尖n. 轰动的演出，巨大的成功v. (cause to) break into pieces violently 打碎，粉碎adulthoodn. 成年aspectn. a particular part or feature of sth. being considered 方面careern. a profession or occupation with opportunities for advancement or promotion 职业;生涯formulan. 公式，程式;准则，方案explorev. 探索;探测;勘探gospeln. (= gospel music) 福音音乐(美国黑人的一种宗教音乐，具有爵士音乐和美国黑人伤感歌曲色彩)jazzn. 爵士音乐rhythmn. 节奏;韵律*albumn. 1. a long-playing record with several items by the same performer (同一表演者的)集锦密纹唱片2. a book with blank pages for stamps, photographs, etc. 集邮册，相册maturea. fully grown or developed mentally or physically 充分发育的;(智力或体力)成熟的independenta. 独立的，自主的tragedyn. 1. a terrible event that causes great sadness 惨事，灾变2. a serious play with a sad ending 悲剧involvevt. 使陷入，使卷入;牵扯，连累coman. 昏迷musicala. of or for music 音乐的geniusn. 天才;创造能力;天才人物conquervt. gain control over (sth. unfriendly or difficult) 征服;克服(困难等)performancen. the acting of a play, the playing of a piece of music, the doing of a dance, etc., in front of an audience 演出，表演，演奏reevaluatevt. 重新评价goaln. 1. an end; objective 目的;目标2.(足球等的)球门;得分进球hungern. state of not having enough to eat; lack of food 饥饿urgevt. 力劝;恳求;敦促raciala. characteristic of race; due to or resulting from race 种族的;由种族引起的harmonyn. agreement (of feelings, interests, opinions, etc.) 和睦，融洽，一致apartheidn. (南非的)种族隔离famen. the condition of being known or talked about a lot 名声，名望activityn. 活动;行动Phrases and Exgressionsbreak intobegin suddenly (to cry, sing, laugh, etc.) 突然(哭、唱、笑)起来bring uptake care of during infancy and childhood; nurse and educate 抚养;养育as far asto the degree that 就…;尽…;至于get tired ofbe no longer interested in 厌倦，厌烦wear outmake useless by use 把…用坏;把…穿破with easewithout difficulty 容易地，无困难地grow intobecome gradually with the passage of time 成长的congratulate oneself on /that ...因…而暗自庆幸Proper NamesStevie Wonder史蒂威·旺达(人名)Morris莫里斯(姓氏)Ronnie White罗尼·怀特(人名)Motown Records莫顿唱片公司Innervisions《内心幻觉》(唱片名)Martin Luther King, Jr.小马丁·路德·金(1929 — 1968，美国民权运动领袖) Nelson Mandela纳尔逊·曼德拉(1918 —，南非共和国总统)。

Theory and Applications

3 A Common Theory Framework
The goal in papers 1 and 2 is to solve The Optimization Problem. Compute MG A; B = ming2G H A; gB. A closely related problem is The Decision Problem. Given 0, does there exist g 2 G such that H A; gB ? The smallest = for which the answer to the decision problem is yes is the answer to the optimization problem. The general strategy used in both theory papers is to develop an algorithm for the decision problem and then use this algorithm in some form of parametric search 13 to get an algorithm for the optimization problem. A common framework is used in attacking the decision problem. This framework turns the decision problem into a geometric intersection problem. For the moment we assume G = T . Let C denote the closed ball" of radius centered at the origin: C = fx 2 Rd : kxk g: 2

Shrinkwrapping and the taming of hyperbolic 3-manifolds

a rX iv:mat h /47161v3[mat h.GT]22O ct25SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC 3–MANIFOLDS DANNY CALEGARI AND DAVID GABAI 0.I NTRODUCTION During the period 1960–1980,Ahlfors,Bers,Kra,Marden,Maskit,Sullivan,Thurston and many others developed the theory of geometrically ﬁnite ends of hy-perbolic 3–manifolds.It remained to understand those ends which are not geo-metrically ﬁnite;such ends are called geometrically inﬁnite .Around 1978William Thurston gave a conjectural description of geometrically inﬁnite ends of complete hyperbolic 3–manifolds.An example of a geometrically inﬁnite end is given by an inﬁnite cyclic covering space of a closed hyperbolic 3-manifold which ﬁbers over the circle.Such an end has cross sections of uniformly bounded area.By contrast,the area of sections of geometrically ﬁnite ends grow exponentially in the distance from the convex core.For the sake of clarity we will assume throughout this introduction that N =H 3/Γwhere Γis parabolic free.Precise statements of the parabolic case will be given in §7.Thurston’s idea was formalized by Bonahon [Bo]and Canary [Ca]with the fol-lowing.Deﬁnition 0.1.An end E of a hyperbolic 3-manifold N is simply degenerate if it has a closed neighborhood of the form S ×[0,∞)where S is a closed surface,and there exists a sequence {S i }of CAT (−1)surfaces exiting E which are homotopic to S ×0in E .This means that there exists a sequence of maps f i :S →N such that the induced path metrics induce CAT (−1)structures on the S i ’s,f (S i )⊂S ×[i,∞)and f is homotopic to a homeomorphism onto S ×0via a homotopy supported in S ×[0,∞).Here by CAT (−1),we mean as usual a geodesic metric space for which geo-desic triangles are “thinner”than comparison triangles in hyperbolic space.If themetrics pulled back by the f i are smooth,this is equivalent to the condition that the Riemannian curvature is bounded above by −1.See [BH]for a reference.Note that by Gauss–Bonnet,the area of a CAT (−1)surface can be estimated from its Euler characteristic;it follows that a simply degenerate end has cross sections of uniformly bounded area,just like the end of a cyclic cover of a manifold ﬁbering over the circle.Francis Bonahon [Bo]observed that geometrically inﬁnite ends are exactly those ends possessing an exiting sequence of closed geodesics.This will be our working deﬁnition of such ends throughout this paper.2DANNY CALEGARI AND DAVID GABAIThe following is our main result.Theorem0.2.An end E of a complete hyperbolic3-manifold N withﬁnitely generated fundamental group is simply degenerate if there exists a sequence of closed geodesics exit-ing E.Consequently we have,Theorem0.3.Let N be a complete hyperbolic3-manifold withﬁnitely generated funda-mental group.Then every end of N is geometrically tame,i.e.it is either geometrically ﬁnite or simply degenerate.In1974Marden[Ma]showed that a geometricallyﬁnite hyperbolic3-manifold is topologically tame,i.e.is the interior of a compact3-manifold.He asked whether all complete hyperbolic3-manifolds withﬁnitely generated fundamental group are topologically tame.This question is now known as the Tame Ends Conjecture or Marden Conjecture.Theorem0.4.If N is a complete hyperbolic3-manifold withﬁnitely generated funda-mental group,then N is topologically tame.Ian Agol[Ag]has independently proven Theorem0.4.There have been many important steps towards Theorem0.2.The seminal re-sult was obtained by Thurston([T],Theorem9.2)who proved Theorems0.3and 0.4for certain algebraic limits of quasi Fuchsian groups.Bonahon[Bo]estab-lished Theorems0.2and0.4whenπ1(N)is freely indecomposible and Canary [Ca]proved that topological tameness implies geometrical tameness.Results in the direction of0.4were also obtained by Canary-Minsky[CaM],Kleineidam–Souto[KS],Evans[Ev],Brock–Bromberg–Evans–Souto[BBES],Ohshika,Brock–Souto[BS]and Souto[So].Thurstonﬁrst discovered how to obtain analytic conclusions from the existence of exiting sequences of CAT(−1)surfaces.Thurston’s work as generalized by Bonahon[Bo]and Canary[Ca]combined with Theorem0.2yields a positive proof of the Ahlfors’Measure Conjecture[A2].Theorem0.5.IfΓis aﬁnitely generated Kleinian group,then the limit set LΓis either S2∞or has Lebesgue measure zero.If LΓ=S2∞,thenΓacts ergodically on S2∞.Theorem0.5is one of the many analytical consequences of our main result. Indeed Theorem0.2implies that a complete hyperbolic3-manifold N withﬁnitely generated fundamental group is analytically tame as deﬁned by Canary[Ca].It follows from Canary that the various results of§9[Ca]hold for N.Our main result is the last step needed to prove the following monumental result,the other parts being established by Alhfors,Bers,Kra,Marden,Maskit, Mostow,Prasad,Sullivan,Thurston,Minsky,Masur–Minsky,Brock–Canary–Minsky, Ohshika,Kleineidam–Souto,Lecuire,Kim–Lecuire–Ohshika,Hossein–Souto and Rees.See[Mi]and[BCM].Theorem0.6(Classiﬁcation Theorem).If N is a complete hyperbolic3-manifold with ﬁnitely generated fundamental group,then N is determined up to isometry by its topolog-ical type,the conformal boundary of its geometricallyﬁnite ends and the ending lamina-tions of its geometrically inﬁnite ends.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS3 The following result was conjectured by Bers,Sullivan and Thurston.Theorem0.4is one of many results,many of them recent,needed to build a proof.Major contributions were made by Alhfors,Bers,Kra,Marden,Maskit,Mostow,Prasad, Sullivan,Thurston,Minsky,Masur–Minsky,Brock–Canary–Minsky,Ohshika,Kleineidam–Souto,Lecuire,Kim–Lecuire–Ohshika,Hossein–Souto,Rees,Bromberg and Brock–Bromberg.Theorem0.7(Density Theorem).If N=H3/Γis a completeﬁnitely generated3-manifold withﬁnitely generated fundamental group,thenΓis the algebraic limit of geo-metricallyﬁnite Kleinian groups.The main technical innovation of this paper is a new technique called shrinkwrap-ping for producing CAT(−1)surfaces in hyperbolic3-manifolds.Historically,such surfaces have been immensely important in the study of hyperbolic3-manifolds,e.g.see[T],[Bo],[Ca]and[CaM].Given a locallyﬁnite set∆of pairwise disjoint simple closed curves in the3-manifold N,we say that the embedded surface S⊂N is2-incompressible rel.∆ifevery compressing disc for S meets∆at least twice.Here is a sample theorem. Theorem0.8(Existence of shrinkwrapped surface).Let M be a complete,orientable, parabolic free hyperbolic3–manifold,and letΓbe aﬁnite collection of pairwise disjoint sim-ple closed geodesics in M.Further,let S⊂M\Γbe a closed embedded2–incompressible surface rel.Γwhich is either nonseparating in M or separates some component ofΓfrom another.Then S is homotopic to a CAT(−1)surface T via a homotopyF:S×[0,1]→Msuch that(1)F(S×0)=S(2)F(S×t)=S t is an embedding disjoint fromΓfor0≤t<1(3)F(S×1)=T(4)If T′is any other surface with these properties,then area(T)≤area(T′)We say that T is obtained from S by shrinkwrapping rel.Γ,or ifΓis understood,T is obtained from S by shrinkwrapping.In fact,we prove the stronger result that T isΓ–minimal(to be deﬁned in§1)which implies in particular that it is intrinsically CAT(−1)Here is the main technical result of this paper.Theorem0.9.Let E be an end of the complete orientable hyperbolic3-manifold N withﬁnitely generated fundamental group.Let C be a3-dimensional compact core of N,∂E Cthe component of∂C facing E and g=genus(∂E C).If there exists a sequence of closed geodesics exiting E,then there exists a sequence{S i}of CAT(−1)surfaces of genus g exit-ing E such that each{S i}is homologically separating in E.That is,each S i homologically separates∂E C from E.Theorem0.4can now be deduced from Theorem0.9and Souto[So];however,we prove that Theorem0.9implies Theorem0.4using only3-manifold topologyand elementary hyperbolic geometry.The proof of Theorem0.9blends elementary aspects of minimal surface theory, hyperbolic geometry,and3-manifold topology.The method will be demonstrated4DANNY CALEGARI AND DAVID GABAIin§4where we give a proof of Canary’s theorem.Theﬁrst time reader is urged to begin with that section.This paper is organized as follows.In§1and§2we establish the shrinkwrap-ping technique forﬁnding CAT(−1)surfaces in hyperbolic3-manifolds.In§3we prove the existence ofǫ-separated simple geodesics exiting the end of parabolic free manifolds.In§4we prove Canary’s theorem.This proof will model the proof of the general case.The general strategy will be outlined at the end of that section. In§5we develop the topological theory of end reductions in3-manifolds.In§6we give the proofs of our main results.In§7we give the necessary embellishments of our methods to state and prove our results in the case of manifolds with parabolic cusps.Notation0.10.If X⊂Y,then N(X)denotes a regular neighborhood of X in Y and int(X)denotes the interior of X.If X is a topological space,then|X|denotes the number of components of X.If A,B are topological subspaces of a third space, then A\B denotes the intersection of A with the complement of B. Acknowledgements0.11.Theﬁrst author is grateful to Nick Makarov for some useful analytic discussions.The second author is grateful to Michael Freedman for many long conversations in Fall1996which introduced him to the Tame Ends con-jecture.He thanks Francis Bonahon,Yair Minsky and Jeff Brock for their interest and helpful comments.Part of this research was carried out while he was visiting Nara Women’s University,the Technion and the Institute for Advanced Study.He thanks them for their hospitality.We thank the referees for their many thoughtful suggestions and comments.1.S HRINKWRAPPINGIn this section,we introduce a new technical tool forﬁnding CAT(−1)surfaces in hyperbolic3–manifolds,called shrinkwrapping.Roughly speaking,given a col-lection of simple closed geodesicsΓin a hyperbolic3–manifold M and an embed-ded surface S⊂M\Γ,a surface T⊂M is obtained from S by shrinkwrapping S rel.Γif it homotopic to S,can be approximated by an isotopy from S supported in M\Γ,and is least area subject to these constraints.Given mild topological conditions on M,Γ,S(namely2–incompressibility,to be deﬁned below)the shrinkwrapped surface exists,and is CAT(−1)with respect to the path metric induced by the Riemannian metric on M.We use some basic analytical tools throughout this section,including the Gauss–Bonnet formula,the coarea formula,and the Arzela–Ascoli theorem.At a number of points we must invoke results from the literature to establish existence of min-imal surfaces([MSY]),existence of limits with area and curvature control([CiSc]), and regularity of the shrinkwrapped surfaces alongΓ([Ri],[Fre]).General refer-ences are[CM],[Js][Fed]and[B].1.1.Geometry of surfaces.For convenience,we state some elementary but fun-damental lemmas concerning curvature of(smooth)surfaces in Riemannian3-manifolds.We use the following standard terms to refer to different kinds of minimal sur-faces:Deﬁnition1.1.A smooth surfaceΣin a Riemannian3-manifold is minimal if it is a critical point for area with respect to all smooth compactly supported variations.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS5 It is locally least area(also called stable)if it is a local minimum for area with respect to all smooth,compactly supported variations.A closed,embedded surface is globally least area if it is an absolute minimum for area amongst all smooth surfaces in its isotopy class.Note that we do not require that our minimal or locally least area surfaces are complete.Any subsurface of a globally least area surface is locally least area,and a locally least area surface is minimal.A smooth surface is minimal iff its mean curvature vectorﬁeld vanishes identically.For more details,consult[CM],especially chapter 5.The intrinsic curvature of a minimal surface is controlled by the geometry of the ambient manifold.The following lemma is formula5.6on page100of[CM]. Lemma1.2(Monotonicity of curvature).LetΣbe a minimal surface in a Riemannian manifold M.Let KΣdenote the curvature ofΣ,and K M the sectional curvature of M. Then restricted to the tangent space TΣ,1KΣ=K M−6DANNY CALEGARI AND DAVID GABAIfor smallǫ,whereφ(·,0)=Id|∂Σ,andφ(∂Σ,t)for small t is the boundary inΣof the tubular t neighborhood of∂Σ.Then∂Σκdl=−da1a3in the complete simply–connected Riemannian2–manifold of con-stant sectional curvatureκ,where the edges a i a j and a j satisfylength(a i a j)=length(a j)Given a point x∈a1a2on one of the edges of a1a2a3,there is a corresponding point a1a1xa toSHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS7 Deﬁnition1.8(Γ–minimal surfaces).Letκ∈R be given.Let M be a complete Riemannian3–manifold with sectional curvature bounded above byκ,and letΓbe an embedded collection of simple closed geodesics in M.An immersionψ:S→MisΓ–minimal if it is smooth with mean curvature0in M\Γ,and is metrically CAT(κ)with respect to the path metric induced byψfrom the Riemannian metric on M.Notice by Lemma1.2that a smooth surface S with mean curvature0in M is CAT(κ),so a minimal surface(in the usual sense)is an example of aΓ–minimal surface.1.3.Statement of shrinkwrapping theorem.Deﬁnition1.9(2–incompressibility).An embedded surface S in a3–manifold M disjoint from a collectionΓof simple closed curves is said to be2–incompressible rel.Γif any essential compressing disk for S must intersectΓin at least two points.If Γis understood,we say S is2–incompressible.Theorem1.10(Existence of shrinkwrapped surface).Let M be a complete,orientable, parabolic free hyperbolic3–manifold,and letΓbe aﬁnite collection of pairwise disjoint sim-ple closed geodesics in M.Further,let S⊂M\Γbe a closed embedded2–incompressible surface rel.Γwhich is either nonseparating in M or separates some component ofΓfrom another.Then S is homotopic to aΓ–minimal surface T via a homotopyF:S×[0,1]→Msuch that(1)F(S×0)=S(2)F(S×t)=S t is an embedding disjoint fromΓfor0≤t<1(3)F(S×1)=T(4)If T′is any other surface with these properties,then area(T)≤area(T′)We say that T is obtained from S by shrinkwrapping rel.Γ,or ifΓis understood,T is obtained from S by shrinkwrapping.The remainder of this section will be taken up with the proof of Theorem1.10. Remark1.11.In fact,for our applications,the property we want to use of our surface T is that we can estimate its diameter(rel.the thin part of M)from its Euler characteristic.This follows from a Gauss–Bonnet estimate and the bounded diameter lemma(Lemma1.15,to be proved below).In fact,our argument will show directly that the surface T satisﬁes Gauss–Bonnet;the fact that it is CAT(−1) is logically superﬂuous for the purposes of this paper.1.4.Deforming metrics along geodesics.Deﬁnition1.12(δ–separation).LetΓbe a collection of disjoint simple geodesics in a Riemannian manifold M.The collectionΓisδ–separated if any pathα:I→M with endpoints onΓand satisfyinglength(α(I))≤δ8DANNY CALEGARI AND DAVID GABAIis homotopic rel.endpoints intoΓ.The supremum of suchδis called the separation constant ofΓ.The collectionΓis weaklyδ–separated ifdist(γ,γ′)>δwheneverγ,γ′are distinct components ofΓ.The supremum of suchδis called the weak separation constant ofΓ.Deﬁnition1.13(Neighborhood and tube neighborhood).Let r>0be given.For a point x∈M,we let N r(x)denote the closed ball of radius r about x,and let N<r(x),∂N r(x)denote respectively the interior and the boundary of N r(x).For a closed geodesicγin M,we let N r(γ)denote the closed tube of radius r aboutγ, and let N<r(γ),∂N r(γ)denote respectively the interior and the boundary of N r(γ). IfΓdenotes a union of geodesicsγi,then we use the shorthand notationN r(Γ)= γi N r(γi)Remark1.14.Topologically,∂N r(x)is a sphere and∂N r(γ)is a torus,for sufﬁ-ciently small r.Similarly,N r(x)is a closed ball,and N r(γ)is a closed solid torus. IfΓisδ–separated,then Nδ/2(Γ)is a union of solid tori.Lemma1.15(Bounded Diameter Lemma).Let M be a complete hyperbolic3–manifold. LetΓbe a disjoint collection ofδ–separated embedded geodesics.Letǫ>0be a Margulis constant for dimension3,and let M≤ǫdenote the subset of M where the injectivity radius is at mostǫ.If S⊂M\Γis a2–incompressibleΓ–minimal surface,then there is a con-stant C=C(χ(S),ǫ,δ)∈R and n=n(χ(S),ǫ,δ)∈Z such that for each component S i of S∩(M\M≤ǫ),we havediam(S i)≤CFurthermore,S can only intersect at most n components of M≤ǫ.Proof.Since S is2–incompressible,any point x∈S either lies in M≤ǫ,or is the center of an embedded m–disk in S,wherem=min(ǫ/2,δ/2)Since S is CAT(−1),Gauss–Bonnet implies that the area of an embedded m–disk in S has area at least2π(cosh(m)−1)>πm2.This implies that if x∈S∩M\M≤ǫthenarea(S∩N m(x))≥πm2The proof now follows by a standard covering argument.A surface S satisfying the conclusion of the Bounded Diameter Lemma is some-times said to have diameter bounded by C modulo M≤ǫ.Remark1.16.Note that ifǫis a Margulis constant,then M≤ǫconsists of Margulis tubes and cusps.Note that the same argument shows that,away from the thin part of M and anǫneighborhood ofΓ,the diameter of S can be bounded by a constant depending only onχ(S)andǫ.The basic idea in the proof of Theorem1.10is to search for a least area repre-sentative of the isotopy class of the surface S,subject to the constraint that the track of this isotopy does not crossΓ.Unfortunately,M\Γis not complete,so the prospects for doing minimal surface theory in this manifold are remote.To rem-edy this,we deform the metric on M in a neighborhood ofΓin such a way thatSHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS9 we can guarantee the existence of a least area surface representative with respect to the deformed metric,and then take a limit of such surfaces under a sequence of smaller and smaller such metric deformations.We describe the deformations of interest below.In fact,for technical reasons which will become apparent in§1.8,the defor-mations described below are not quite adequate for our purposes,and we must consider metrics which are deformed twice—ﬁrstly,a mild deformation which satisﬁes curvature pinching−1≤K≤0,and which is totally Euclidean in a neighborhood ofΓ,and secondly a deformation analogous to the kind described below in Deﬁnition1.17,which is supported in this totally Euclidean neighbor-hood.Since the reason for this“double perturbation”will not be apparent until §1.8,we postpone discussion of such deformations until that time.Deﬁnition1.17(Deforming metrics).Letδ>0be such thatΓisδ–separated. Choose some small r with r<δ/2.For t∈[0,1)we deﬁne a family of Riemannian metrics g t on M in the following manner.The metrics g t agree with the hyperbolic metric away from someﬁxed tubular neighborhood N r(Γ).Leth:N r(1−t)(Γ)→[0,r(1−t)]be the function whose value at a point p is the hyperbolic distance from p to Γ.We deﬁne a metric g t on M which agrees with the hyperbolic metric outside N r(1−t)(Γ),and on N r(1−t)(Γ)is conformally equivalent to the hyperbolic metric, as follows.Letφ:[0,1]→[0,1]be a C∞bump function,which is equal to1on the interval[1/3,2/3],which is equal to0on the intervals[0,1/4]and[3/4,1],and which is strictly increasing on[1/4,1/3]and strictly decreasing on[2/3,3/4].Then deﬁne the ratiog t length elementr(1−t) We are really only interested in the behaviour of the metrics g t as t→1.As such,the choice of r is irrelevant.However,for convenience,we willﬁx some small r throughout the remainder of§1.The deformed metrics g t have the following properties:Lemma1.18(Metric properties).The g t metric satisﬁes the following properties:(1)For each t there is an f(t)satisfying r(1−t)/4<f(t)<3r(1−t)/4such thatthe union of tori∂N f(t)(Γ)are totally geodesic for the g t metric(2)For each componentγi and each t,the metric g t restricted to N r(γi)admits afamily of isometries which preserveγi and acts transitively on the unit normal bundle(in M)toγi(3)The area of a disk cross–section on N r(1−t)is O((1−t)2).(4)The metric g t dominates the hyperbolic metric on2–planes.That is,for all2–vectorsν,the g t area ofνis at least as large as the hyperbolic area ofνProof.Statement(2)follows from the fact that the deﬁnition of g t has the desired symmetries.Statements(3)and(4)follow from the fact that the ratio of the g t metric to the hyperbolic metric is pinched between1and3.Now,a radially sym-metric circle linkingΓof radius s has length2πcosh(s)in the hyperbolic metric, and therefore has length2πcosh(s)(1+2φ(s/r(1−t)))10DANNY CALEGARI AND DAVID GABAIin the g t metric.For sufﬁciently small(butﬁxed)r,this function of s has a local minimum on the interval[r(1−t)/4,3r(1−t)/4].It follows that the family of radially symmetric tori linking a component ofΓhas a local minimum for area in the interval[r(1−t)/4,3r(1−t)/4].By property(2),such a torus must be totally geodesic for the g t metric. Notation1.19.We denote length of an arcα:I→M with respect to the g t metricas lengtht (α(I)),and area of a surfaceψ:R→M with respect to the g t metric asarea t(ψ(R)).1.5.Constructing the homotopy.As aﬁrst approximation,we wish to construct surfaces in M\Γwhich are globally least area with respect to the g t metric.There are various tools for constructing least area surfaces in Riemannian3-manifolds under various conditions,and subject to various constraints.Typically one works in closed3-manifolds,but if one wants to work in3-manifolds with boundary,the “correct”boundary condition to impose is mean convexity.A co-oriented surface in a Riemannian3-manifold is said to be mean convex if the mean curvature vec-tor of the surface always points to the negative side of the surface,where it does not vanish.Totally geodesic surfaces and other minimal surfaces are examples of mean convex surfaces,with respect to any co-orientation.Such surfaces act as bar-riers for minimal surfaces,in the following sense:suppose that S1is a mean convex surface,and S2is a minimal surface.Suppose further that S2is on the negative side of S1.Then if S2and S1are tangent,they are equal.One should stress that this barrier property is local.See[MSY]for a more thorough discussion of barrier surfaces.Lemma1.20(Minimal surface exists).Let M,Γ,S be as in the statement of Theo-rem1.10.Let f(t)be as in Lemma1.18,so that∂N f(t)(Γ)is totally geodesic with re-spect to the g t metric.Then for each t,there exists an embedded surface S t isotopic in M\N f(t)(Γ)to S,and which is globally g t–least area among all such surfaces.Proof.Note that with respect to the g t metrics,the surfaces∂N f(t)(Γ)described in Lemma1.18are totally geodesic,and therefore act as barrier surfaces.We remove the tubular neighborhoods ofΓbounded by these totally geodesic surfaces,and denote the result M\N f(t)(Γ)by M′throughout the remainder of this proof.We assume,after a small isotopy if necessary,that S does not intersect N f(t)for any t,and therefore we can(and do)think of S as a surface in M′.Notice that M′is a complete Riemannian manifold with totally geodesic boundary.We will construct the surface S t in M′,in the same isotopy class as S(also in M′).If there exists a lower bound on the injectivity radius in M′with respect to the g t metric,then the main theorem of[MSY]implies that either such a globally least area surface S t can be found,or S is the boundary of a twisted I–bundle over a closed surface in M′,or else S can be homotoped off every compact set in M′.First we show that these last two possibilities cannot occur.If S is nonseparating in M,then it intersects some essential loopβwith algebraic intersection number 1.It follows that S cannot be homotoped offβ,and does not bound an I–bundle. Similarly,ifγ1,γ2are distinct geodesics ofΓseparated from each other by S,then theγi’s can be joined by an arcαwhich has algebraic intersection number1with the surface S.The same is true of any S′homotopic to S;it follows that S cannot be homotoped off the arcα,nor does it bound an I–bundle disjoint fromΓ,and therefore does not bound an I–bundle in M′.SHRINKWRAPPING AND THE TAMING OF HYPERBOLIC3–MANIFOLDS11 Now suppose that the injectivity radius on M′is not bounded below.We use the following trick.Let g′t be obtained from the metric g t by perturbing it on the complement of some enormous compact region E so that it has aﬂaring end there, and such that there is a barrier g′t-minimal surface close to∂E,separating the com-plement of E in M′from S.Then by[MSY]there is a globally g′t least area surface S′t,contained in the compact subset of M′bounded by this barrier surface.Since S′t must either intersectβorα,by the Bounded Diameter Lemma1.15,unless the hyperbolic area of S′t∩E is very large,the diameter of S′t in E is much smaller than the distance fromαorβto∂E.Since by hypothesis,S′t is least area for the g′t metric,its restriction to E has hyperbolic area less than the hyperbolic area of S, and therefore there is an a priori upper bound on its diameter in E.By choosing E big enough,we see that S′t is contained in the interior of E,where g t and g′t agree. Thus S′t is globally least area for the g t metric in M′,and therefore S t=S′t exists for any t.The bounded diameter lemma easily implies the following:Lemma1.21(Compact set).There is aﬁxed compact set E⊂M such that the surfaces S t constructed in Lemma1.20are all contained in E.Proof.Since the hyperbolic areas of the S t are all uniformly bounded(by e.g. the hyperbolic area of S)and are2–incompressible rel.Γ,they have uniformly bounded diameter away fromΓoutside of Margulis tubes.Since for homological reasons they must intersect the compact setsαorβ,they can intersect at most ﬁnitely many Margulis tubes.It follows that they are all contained in aﬁxed bounded neighborhood E ofαorβ,containingΓ.To extract good limits of sequences of minimal surfaces,one generally needs a priori bounds on the area and the total curvature of the limiting surfaces.Here for a surface S,the total curvature of S is just the integral of the absolute value of the(Gauss)curvature over S.For minimal surfaces of aﬁxed topological type in a manifold with sectional curvature bounded above,a curvature bound follows from an area bound by Gauss–Bonnet.However,our surfaces S t are minimal with respect to the g t metrics,which have no uniform upper bound on their sectional curvature,so we must work slightly harder to show that the the S t have uniformly bounded total curvature.More precisely,we show that their restrictions to the complement of anyﬁxed tubular neighborhood Nǫ(Γ)have uniformly bounded total curvature.Lemma1.22(Finite total curvature).Let S t be the surfaces constructed in Lemma1.20. Fix some small,positiveǫ.Then the subsurfacesS′t:=S t∩M\Nǫ(Γ)have uniformly bounded total curvature.Proof.Having chosenǫ,we choose t large enough so that r(1−t)<ǫ/2.Observeﬁrstly that each S t has g t area less that the g t area of S,and therefore hyperbolic area less that the hyperbolic area of S for sufﬁciently large t.Letτt,s=S t∩∂N s(Γ)for small s.By the coarea formula(see[Fed],[CM]page 8)we can estimatearea(S t∩(Nǫ(Γ)\Nǫ/2(Γ)))≥ ǫǫ/2length(τt,s)ds12DANNY CALEGARI AND DAVID GABAIIf the integral of geodesic curvature along a componentσofτt,ǫis large,then the length of the curves obtained by isotopingσinto S t∩Nǫ(Γ)grows very rapidly, by the deﬁnition of geodesic curvature.Since there is an a priori bound on the hyperbolic area of S t,it follows that there cannot be any long components ofτt,s with big integral geodesic curvature.More precisely,consider a long componentσofτt,s.For l∈[0,ǫ/2]the boundaryσl of the l-neighborhood ofσin S t∩Nǫ(Γ)is contained in Nǫ(Γ)\Nǫ/2(Γ).If the integral of the geodesic curvature alongσl were sufﬁciently large for every l,then the derivative of the length of theσl would be large for every l,and therefore the lengths of theσl would be large for all l∈[ǫ/4,ǫ/2].It follows that the hyperbolic area of theǫ/2collar neighborhood ofσin S t would be very large,contrary to existence of an a priori upper bound on the total hyperbolic area of S t.This contradiction implies that for some l,the integral of the geodesic curvature alongσl can be bounded from above.To summarize,for each constant C1>0 there is a constant C2>0,such that for each componentσofτt,ǫwhich has length ≥C1there is a loopσ′⊂S t∩(Nǫ(Γ)\Nǫ/2(Γ))isotopic toσby a short isotopy,satisfyingσ′κdl≤C2On the other hand,since S t is g t minimal,there is a constant C1>0such that each componentσofτt,ǫwhich has length≤C1bounds a hyperbolic globally least area disk which is contained in M\Nǫ/2(Γ).For t sufﬁciently close to1,such a disk is contained in M\N r(1−t)(Γ),and therefore must actually be a subdisk of S t.By the coarea formula above,we can chooseǫso that length(τt,s)is a priori bounded.It follows that if S′′t is the subsurface of S t bounded by the compo-nents ofτt,s of length>C1then we have a priori upper bounds on the area of S′′t, on ∂S′′tκdl,and on−χ(S′′t).Moreover,S′′t is contained in M\N r(1−t)where the metric g t agrees with the hyperbolic metric,so the curvature K of S′′t is bounded above by−1pointwise,by Lemma1.2.By the Gauss–Bonnet formula,this gives an a priori upper bound on the total curvature of S′′t,and therefore on S′t⊂S′′t. Remark1.23.A more highbrow proof of Lemma1.22follows from Theorem1 of[S],using the fact that the surfaces S′t are locally least area for the hyperbolic metric,for t sufﬁciently close to1(depending onǫ).Lemma1.24(Limit exists).Let S t be the surfaces constructed in Lemma1.20.Then there is an increasing sequence0<t1<t2<···such that lim i→∞t i=1,and the S tconverge on compact subsets of M\Γin the C∞itopology to some T′⊂M\Γwith closure T in M.Proof.By deﬁnition,the surfaces S t have g t area bounded above by the g t area of S.Moreover,since S is disjoint fromΓ,for sufﬁciently large t,the g t area of S is equal to the hyperbolic area of S.Since the g t area dominates the hyperbolic area, it follows that the S t have hyperbolic area bounded above,and by Lemma1.22, for anyǫ,the restrictions of S t to M\Nǫ(Γ)have uniformly boundedﬁnite total curvature.。

Group laws and free subgroups in topological groups

a r X i v:mat h /36364v1[mat h.GR]25J u n23Group laws and free subgroups in topological groups Mikl´o s Ab´e rt June 25,2003Abstract We prove that a permutation group in which diﬀerent ﬁnite sets have diﬀerent stabilizers cannot satisfy any group law.For locally compact topological groups with this property we show that almost all ﬁnite subsets of the group generate free subgroups.We derive consequences of these theorems on Thompson’s group F ,weakly branch groups,automorphism groups of regular trees and proﬁnite groups with alternating composition factors of unbounded degree.1Introduction Let G be a permutation group acting on a set X .We say that G separates X if for any ﬁnite subset Y ⊆X the pointwise stabilizer G Y does not stabilize any point outside Y .A group G satisﬁes a group law,if there exists a word w ∈F k such that w (g 1,g 2,...,g k )=1for all g i ∈G .Our ﬁrst result is the following.Theorem 1If G separates X then G does not satisfy any group law.In contrast,for the natural transitive permutation action of the so-called lamplighter group L (the wreath product C 2≀Z )the stabilizer of n points stabi-lizes at most n other points.On the other hand,L is metabelian,so it satisﬁes a commutator identity.Theorem 1allows us to give a short proof of the main result in [BM].Thomp-son’s group F is the group of piecewise linear orientation-preserving homeomor-phisms of the unit interval with dyadic breaking points and slopes which are powers of 2.Corollary 2Thompson’s group F does not satisfy any group law.Note that by [BM]F does not contain any free subgroups.We can generalise Theorem 1for locally compact topological groups which act on a set separating it and show that generic subgroups of them are free.Altough we derive the compact case as a simple consequence of the general result1on locally compact groups,we state it diﬀerently as the necessary restrictions are more natural.Let G be a compact topological group.Then there is a unique G-invariant probability measureµon G.This extends to a probability measure on G n.A subset of G n is almost sure(a.s.)if its complement has zero probability.An action of G on a set X is topological,if for all x∈X the point stabilizer G x is closed.If G is also transitive,then X can be identiﬁed with the coset space G/H for any point stabilizer H and as such,it carries a natural G-invariant measureλ.A groupΓ⊆Sym(Ω)acts freely onΩif for every x∈Ωthe point-stabilizer Γx is trivial.A group H⊆G acts almost freely on X if there is Y⊆X such thatλ(X\Y)=0,Y isΓ-invariant andΓacts freely on Y.Theorem3Let G be a compact topological group acting topologically on X and separating X.Then n random elements in G generate a free subgroup of rank n a.s.If G is transitive then this group acts almost freely on X a.s.So somewhat surprisingly,in the transitive case a genericﬁnitely generated subgroup of G does exactly the opposite of separating X,as almost all points have a stabilizer which stabilizes almost all points in X.Theorem3applies to proﬁnite weakly branch groups.A weakly branch group can be deﬁned as a group acting spherically transitively on a rooted tree such that the rigid stabilizer of every vertex is nontrivial.Recall that the rigid stabilizer of a vertex v is the set of group elements which move only descendants of v.The class of weakly branch groups contains many groups with interesting properties.Examples are theﬁrst Grigorchuk group([Gri]),Wilson’s group of non-uniform exponential growth([Wi1]),Segal’s group of slow subgroup growth ([Seg])and the group constructed by Grigorchuk and Zuk which is not in the class SG([GZ])but as Bartholdi and Vir´a g have recently shown is amenable ([BV]).A proﬁnite weakly branch group is a weakly branch group which is closed in the proﬁnite topology of the automorphism group of the corresponding tree, e.g.,the closure of any weakly branch group.Corollary4If G is a proﬁnite weakly branch group then n random elements generate a free subgroup of rank n in G a.s.Moreover,if the underlying tree has bounded degree and G isﬁnitely generated then G contains aﬁnitely generated dense free subgroup.The above result generalises a result of Wilson[Wi2]who proved that for branch proﬁnite groups the subset of n-tuples in G n generating non-free sub-groups is meagre(ﬁrst category)in G n.His result implies thatﬁnitely generated pro-p branch groups haveﬁnitely generated dense free subgroups,but for the general proﬁnite setting our stronger probabilistic result is needed.Theorem3also applies to inﬁnite iterated wreath products of nontrivial ﬁnite permutation groups.This reproves a result of Bhattacharjee[Bha]saying that in these groups n elements generate a free subgroup of rank n a.s.2We deduce Theorem3as a special case of a more general theorem on locally compact groups.If the group G is locally compact but not compact,we do not have a G-invariant probability measure anymore.Also,we have to strengthen the notion of topological action,since a countable group can easily act on a set, separating it,without containing any free subgroups.Thompson’s group or the ﬁnitary symmetric group acting on N both have this property.A locally compact group G carries a right G-invariant Haar measureµ.An action of G on a set X is topological,if for all x∈X the point stabilizer G x is closed andµ(G x)=0.The action is strongly topological,if for allﬁnite subset Y⊆X the pointwise stabilizer G Y acts topologically on X\Y.A property of n-tuples of G is almost sure(a.s.)if its complement isµ-negligible in G n.Note that these notions are independent of the choice ofµ.If G is transitive,then X again can be identiﬁed with the coset space G/H for some closed subgroup H and as such,it inherits a natural measureλfrom G.This measure is not necessarily G-invariant,but always G-equivariant.Theorem5Let G be a locally compact topological group acting strongly topo-logically on X and separating X.Then n elements in G generate a free subgroup of rank n a.s.If G is transitive then this group acts almost freely on X a.s.A natural class of locally compact groups to which Theorem5applies are automorphism groups of unrooted regular trees.Corollary6For any regular tree T,n elements in Aut(T)generate a free subgroup of rank n a.s.This corollary is used in[AG]to show that Aut(T)containsﬁnitely generated dense free subgroups for every unrooted regular tree T.Using an asymptotic version of Theorem3we can also answer a question of Pyber.Dixon,Pyber,Seress and Shalev in[DPSS]gave an elegant new proof to a conjecture of Magnus.The original proof was given by Weigel[Wei]using diﬀerent methods.Magnus conjectured that the free group F2is residually S for every inﬁnite set S of pairwise non-isomorphic nonabelianﬁnite simple groups. Recall that a group G is residually S if the intersection of all normal subgroups with quotients in S is trivial in G.One of the corollaries in[DPSS]is that if a proﬁnite group G has inﬁnitely many nonabelian simple quotients then n random elements in G generate a free subgroup a.s.for all n.It is asked in[DPSS,Problem7]whether one can generalise this to proﬁnite groups with inﬁnitely many nonabelian simple composition factors.Our result makes aﬁrst step into this direction.Theorem7Let G be a proﬁnite group which has alternating composition fac-tors of unbounded degree.Then n random elements generate a free subgroup in G a.s.for all n.32ProofsIn this section we prove the theorems and corollaries from the introduction.Proof of Theorem1.We can assume that G acts transitively on X.If the action is intransitive,let X′be a nontrivial orbit of G.Then X′is inﬁnite, otherwise for every x′∈X′and Y=X′\{x′}the stabilizer G Y would stabilize x′,contradicting the separability.By the same argument for everyﬁnite Y⊆X the orbits of the action of G Y on X\Y are all inﬁnite.Let F k be the free group on k variables f1,f2,...,f k.Let w∈F k be a reduced word of length n,i.e.,w=v1v2···v nwherev i∈ f1,f−11,...,f k,f−1kwith v i v i+1=1for1≤i≤n−1.Let w j be the j-th beginning segment of w,w j=v1v2···v j(0≤j≤n).For a k-tuple(g1,g2,...,g k)∈G k we denote w j(g1,g2,...,g k)∈G by the value of the word w j in G via the substitution f i=g i.We claim that for all1=w∈F k and for all x0∈X there exist a k-tuple (h1,h2,...,h k)∈G k such that the points(0≤j≤n)x j=x w j(h1,h2,...,h k)are disjoint.This clearly shows that G cannot satisfy the group law w,as this particular substitution moves the point x0to x n=x0.We prove our claim by induction on n.For n=1the word consists of one variable and the claim is trivial.Fix x0∈ing the inductional hypothesis,we have(g1,g2,...,g k)∈G k such that x0,x1,...,x n−1are all disjoint.If x n/∈{x0,x1,...,x n−1}then we found the right(g1,g2,...,g k).So assume that x n=x j for some j<n.Now letI= i<n|v i=v n or v i+1=v−1n .Let m be the index of v n,that is v n∈ f m,f−1m .Also let g=v n(g1,g2,...,g k) (thus g is either g m or g−1m).We claim that j/∈I.If v j=v n,then x g j−1= x j=x g n−1which is a contradiction since j<n and all the x i are disjoint for i<n.If v j+1=v−1n then x g j+1=x j=x g n−1which implies j=n−2and thus v n−1=v−1n which contradicts the assumption that the word w is reduced.So our claim holds and j/∈I.Now let Y={x i|i∈I}and let c∈G Y be an arbitrary element of the stabilizer of Y.4Modify the k-tuple(g1,g2,...,g k)to(h1,h2,...,h k)as follows:h i=g j (i=m),h m=g m c if v n=f m and h m=cg m if v n=f−1m.Using theconstruction of I it is straightforward to check that x h i+1i=x i+1for i<n−1 and x h n n−1=x c j.Since x j/∈Y,the orbit of G Y containing x j is inﬁnite so we can choose c such that x c j/∈{x0,x1,...,x n−1}.So the k-tuple(h1,h2,...,h k)shows that our claim holds and so w(g1,g2,...,g k)=1.Proof of Corollary 2.We show that Thompson’s group F separates the unit interval by its natural action.Let Y⊆[0,1]be aﬁnite set and let x∈[0,1]\Y.Let d1,d2∈(0,1)be dyadic numbers such that x∈[d1,d2]but[d1,d2]∩Y=∅.Then the pointwise stabilizer F d1,d2of[0,d1]∪[d2,1]in F ispermutation isomorphic to F by its action on[d1,d2].In particular,there is anelement of F d1,d2which moves the point x.Proof of Theorem3.We need to check that the conditions for Theorem 5hold,that is,the action of G on X is strongly topological.By the same argument as in the proof of Theorem1we see that every G-orbit has to be inﬁnite.So any point-stabilizer G x must have inﬁnite index in G;since G x is closed and G is compact,G x has zero Haar measure,so the action of G is topological.Also,for anyﬁnite set Y⊆X the pointwise stabilizer G Y trivially separates X\Y and in X\Y the point-stabilizers are closed(being the intersection of closed subgroups).The stabilizers G Y are compact,so by repeating the above argument to the action of G Y on X\Y we see that the action of G on X is strongly topological.Proof of Corollary4.Let G act on the rooted tree T spherically transi-tively such that the rigid vertex stabilizers are nontrivial.Let X be the boundary of the tree,i.e.,the set of inﬁnite rays of T.Since G is closed in the proﬁnite topology of Aut(T),for every x∈X the stabilizers G x are closed,so G acts on X topologically.We claim that G separates X.Let y1,...y n,x∈X be pairwise disjoint rays. Choose a level k such that the vertices in the rays y1,...y n,x at the k-th level are all disjoint.Let x0be the vertex of x at level k.Let S be the stabilizer of x0in G and let R be the rigid vertex stabilizer of x0in G.Then S acts spherically transitively on the inﬁnite subtree rooted at x0,since if a and b are both descendants of x0at the same level,then there is g∈G such that a g=b and clearly g must stabilize x0.Now R is normal in S and since it is nontrivial, it cannot stabilize any inﬁnite ray going through x0.In particular,there exists r∈R such that x r=x.On the other hand,r stabilizes every ray not going through x0.This proves our claim.Now using Theorem3we see that n random elements generate a free sub-group of G with probability1.If the degree of T is bounded,then Aut(T)has onlyﬁnitely many diﬀerent simple upper composition factors.Thus the degree of upper alternating sections of G is bounded.Recall that an upper section is a homomorphic image of5an open ing[BPS]this implies that G has polynomial maximal subgroup growth(PMSG),i.e.,the number of maximal subgroups of index n in G is bounded by a polynomial of n.By[MS]for the proﬁnite group G PMSG is equivalent to that G is positivelyﬁnitely generated(PFG),i.e.,that there is a number k such that k random elements generate a dense subgroup of G with positive probability.So for every n≥k we obtain that n random elements of G generate a dense free subgroup of G with positive probability.This completes the proof.Now we will prove Theorem5.We need a straightforward lemma,the proof of which we include for completeness.If G is a locally compact group and H is a closed subgroup endowed with a right Haar measureλ,thenλnaturally extends to every right H-coset C by λ(X)=λ(Xg−1)where g∈C.The measureλ(X)is independent of the choice of g asλis right invariant in H.Let K⊂H be a closed subgroup of measure zero and letµbe right Haar measure on K.Lemma8Let X be aλ-measurable subset of C such that for almost all K-cosets D⊆C we haveµ(D∩X)=0.Thenλ(X)=0.Proof.Fix c0∈C.Then C=Hc0.Deﬁne the subset khc0Y={(k,h)∈K×H|khc0∈X}.Since Y is the preimage of X by a continuous map from K×H to C,it is measurable.Now for all h0∈H theﬁberR(h0)={k∈K|(k,h0)∈Y}=K∩Xc−10h−10and so by the assumption for almost all h0we haveµ(R(h0))=µ(Kh0c0∩X)=0.So by Fubini’s theorem Y has measure zero in K×H.This implies that for almost all k0∈K theﬁberL(k0)={h∈H|(k0,h)∈Y}=k−10Xc−10has measure zero in H.Pick such a ing thatλis left H-equivariant we getλ(Xc−10)=0in H,that is,λ(X)=0.Proof of Theorem5.We will use the notations from the proof of Theorem 1.We claim that we can assume that G is transitive on X.Let X0⊆X be a G-orbit.The restrictionthe closed stabilizers of points in X0.NowG n have zero measure in G n and our claim holds.Let Y1,Y2,...,Y k⊆X and Z1,Z2,...,Z k⊆X beﬁnite ordered subsets such that for all1≤i≤k we have|Y i|=|Z i|and there exists h i such that Y h i i=Z i as ordered subset.Let Y=∪Y i,Z=∪Z i and letA(Y1,Y2,...,Y k,Z1,Z2,...,Z k)={(g1,g2,...,g k)|Y g i i=Z i for1≤i≤k}.Let us denote S i=G Yithe pointwise stabilizer of Y i in G,let C i=S i h i and let S=S1×S2×···×S k.Obviously,A coincides the right S-coset C1×C2×···×C k so it inherits the natural topology and measure from S.Now let x0∈X\Y be aﬁxed point and let w∈F k be a reduced word of length n>0.Call a tuple(g1,g2,...,g k)∈A distinctive,if for the elementsx j=x0w j(g1,g2,...,g n)we have x i/∈Y∪Z(1≤i≤n)and x i=x j(0≤i<j≤n).LetU=U(Y1,Y2,...,Y k,Z1,Z2,...,Z k,x0,w)be the set of distinctive elements of A.Ourﬁrst claim is that U is open in A.Let H=G xbe the stabilizer of x0 in G.For all0≤i<j≤k let us deﬁneV ij={(g1,g2,...,g k)∈A|x i=x j}.Clearly,U=A\ i<j≤n V ijso it is enough to show that the sets V ij are closed.Deﬁne the function F ij: G k→G byF ij(g1,...,g k)=w i(g1,g2,...,g n)w−1j(g1,g2,...,g n).Since F ij is continuous,the pre-image W ij=F−1ij (H)is closed in G n.On theother hand,V ij=A∩W ij so it is closed.Our second claim is that U is a.s.in A.We use induction on n,the length of w.If n=0,then U=A and the claim obviously holds.If n>0,let f be theﬁrst letter of w.By permuting the letters and possiblyswitching f1and f−11(and also Y1and Z1),we can assume that f=f1.Nowlet R be the stabilizer of x0in S1and let D=Rh1×C2×···×C k.Since G acts strongly topologically on X,R is closed of zero measure in S1,thus D is closed of zero measure in A.The setL={(g1,g2,...,g k)∈A|x g10∈Y∪{x0}}7is theﬁnite union of D-cosets so it has zero measure in A.That is,for almost all(g1,g2,...,g k)∈A we have x g10/∈Y∪{x0}.For aﬁxed x′0=x g10consider the new systemY′1=Y1∪{x0},Z′1=Z1∪{x′0}and Y′j=Y j,Z′j=Z j for2≤j≤k.Also let w′=f−11w reduced(delete theﬁrstletter of w).Now w′has length n−1so by induction U′=U(Y′1,...,Z′k,x′0,w′) is a.s.in A′=A(Y′1,...,Z′k).Since x0∈Y′1,distinctive elements of A′will be also distinctive in ing Lemma8we see that U is a.s.in A what we claimed.Now we prove the theorem.Let k≥1,set X1,...,X k+1and Y1...,Y k+1to be all empty sets and let x0∈X.For a word1=v∈F k let us deﬁne w∈F k+1 by w=f k+1vf−1k+1.Then using the above claim we get that the set of distinctive k+1-tuples is a.s.in G k+1.In particular(x g k+10)v(g1,...,g k)g−1k+1=x w(g1,...,g k+1)=x0so(x g k+10)v(g1,...,g k)=x g k+1for such k+ing Fubini’s theorem this implies that for almost all h∈G the set of distinctive k+1-tuples with g k+1=h starting from x h0is a.s. in G k.So assuming the transitivity of G on X for almost all x∈X the value v(g1,...,g k)does notﬁx ing that there are countably many words in F k and Fubini’s theorem again,we get that for almost all k-tuples(g1,...,g k)∈G k for almost all x∈X the value v(g1,...,g k)does notﬁx x for all1=v∈F k. That is,(g1,...,g k)∈G k generates a free subgroup acting almost freely on X a.s.Proof of Corollary6.Let X be the boundary of the tree,i.e.,the set of inﬁnite rays of T and let G=Aut(T).Then G is locally compact,the neigh-bourhood of the identity being pointwise stabilizers of balls.We can assume that vertex stabilizers have Haar measure1.Clearly,for every x∈X the point stabilizer G x is closed.Let Y⊆X be an arbitrary(possibly empty)ﬁnite subset and let x∈X\Y.If|Y|≥3,then G Y is compact as it the stabilizer of the convex hull C of Y in T.Since C has onlyﬁnitely many limit points in X(the set Y),x has a closest point c in C.Then G Y acts transitively on the shadow of c containing x, so the G Y-orbit containing x is inﬁnite(in fact it contains continuously many points).As G Y is compact,this shows that G Y∪{x}has inﬁnite index and thus zero measure in G Y.If|Y|≤2,then G Y acts transitively on X\Y.Fix a point t∈T,let H be the stabilizer of t in G Y and let H x be the stabilizer of x in H.Then H is compact and has countable index in G Y so it has positive measure in G Y.As the convex cone of Y∪{t}has no new boundary points,the x-orbit of H is inﬁnite and so H x has zero measure in H.Since G Y∪{x}is a countable union of cosets of H x,it has zero measure in G Y.8The above argument shows that G separates X and the action is strongly ing Theorem5we see that the corollary holds.Now we prove Theorem7using a general lemma onﬁnite permutation groups.Let G be a permutation group acting on aﬁnite set X.We say that G separates X in order(n,a)if for any subset Y⊆X with|Y|=n the point stabilizer G Y acts on X\Y in a way such that all orbits have size at least a.For a word w∈F k let P G(w)denote the probability that w is not satisﬁed in G. Lemma9Let G be aﬁnite group separating X of order(n,a)and let w∈F k have length n.Then Q(w)≥ 1−na n−1≥ 1−na therandomly chosen x n does not meet x0,x1,...,x n−1,proving our lemma.Proof of Theorem7.Let A=A k be an alternating upper compositionfactor of ing standard technique there exist m and normal subgroups N⊳M⊳G such that M/N≅A m and|G:N|isﬁnite.Factoring out with the centralizer of M/N in G we get that G quotients to aﬁnite group G which embeds as a subgroup of the automorphism group of A m which contains A mand acts transitively on the factors.Now the whole Aut(A m)is isomorphic to the wreath product S k≀S m whichacts naturally on a set X of km points.Since the natural action of A on k pointsseparates it of order(n,k−n)for any n<k−1,the same holds for the action of A m on X.Now G contains A m,so it also separates X of order(n,k−n)for all n<k−1.Now let w∈F k be a word of length n.If a word is not satisﬁed in the quotient group G then it is not satisﬁed for any preimage in G and the image of a uniform random element in G is uniform random in ing Lemma9, we haveP G(w)≥P G(w)≥1−nAcknowledgement.The author is grateful to the‘Secret Seminar’for the inspiring milieu and to Laci Pyber,who asked him whether Theorem7holds. References[AG]M.Ab´e rt,Y.Glasner,Generic subgroups of automorphism groups of trees,preprint[BV]L.Bartholdi, B.Vir´a g,Amenability via random walks,preprint, /abs/math.GR/0305262[Bha]M.Bhattacharjee,The ubiquity of free subgroups in certain inverse limits of groups.J.Algebra172(1995),no.1,134–146.[BPS] A.Borovik,L.Pyber,A.Shalev,Maximal subgroups inﬁnite and proﬁ-nite groups,Trans.Amer.Math.Soc.348(1996),no.9,3745–3761.[BM]M.G.Brin and G.C.Squier,Groups of piecewise linear homeomor-phisms of the real line,Invent.Math.79(1985),no.3,485–498.[DPSS]J.D.Dixon,L.Pyber,A.Seress and A.Shalev,Residual properties of free groups and probabilistic methods,J.Reine Ang.Math.556(2003),159-172.[Gri]R.I.Grigorchuk,Just inﬁnite branch groups,in New horizons in pro-p groups,121–179,Progr.Math.,184,Birkh¨a user Boston,Boston,MA,2000.[GZ]R.I.Grigorchuk,A.Zuk,On a torsion free weakly branch group deﬁned by a three state automaton,Internat J.Algebra Comput.,12(1-2),223-246[MS] A.Mann,A.Shalev,Simple groups,maximal subgroups,and proba-bilistic aspects of proﬁnite groups,Israel J.Math.96(1996),449–468. [Seg] D.Segal,Theﬁnite images ofﬁnitely generated groups,Proc.London Math.Soc.(3)82(2001),no.3,597–613.[Wei]T.Weigel,Residual properties of free groups I-III,J.Algebra160 (1993),no.1,16–41.,Comm.Algebra20(1992),no.5,1395–1425.,Israel J.Math.77(1992),no.1-2,65–81.[Wi1]J.Wilson,On exponential and uniformly exponential growth for groups, preprint,http://www.unige.ch/math/biblio/preprint/2002/growth.ps [Wi2]J.Wilson,On just inﬁnite abstract and proﬁnite groups,in New hori-zons in pro-p groups,181–203,Progr.Math.,184,Birkh¨a user Boston,Boston,MA,2000.10。

美国财产法提纲

What is property? → Dueling conceptions of private propertyPROPERTY LAW IS ABOUT RELATIONS AMONG PERSONS WITH RESPECT TO THINGS •Resources of the world on which society depends for sustenance, satisfaction its needs and desires o How to reward and encourage efforts for productive use of those resources?•Property rights consist of what you can get the state to back you up ono How law defines property interestso How one acquires propertyo How property rights can be sharedo How outside parties’ interests affect th e rights of OsTRESPASS & NUISANCETrespass to LandAny intentional intrusion that deprives another of possession of land, even if only temporarily •Strict liability tort – one who commits an intentional trespass is subject to liability irrespective of whether he thereby causes any harm to any legally protected interest of another•Sometimes necessity is recognized as a defense to an action for trespassJacque v. Steenberg Homes (Wisc, 1997)•Issue: should award of nominal damages for intentional trespass to land support punitive damage award?o Jacques didn’t want the mobile home moved across their propertyo Steenberg felt it was more convenient to move across the Jacques’ land, a lso potential danger to Steenberg employees if used snow-covered road instead•Holding: upholds punitive damage award even when only nominal damages (no actual damages) awarded o Important to deter physical invasions of land (right to exclude) even when conduct is harmless ▪Infringement of a legal right = always an actual harm, whether measurable or not▪Private LO and society have more than a nominal interest in excluding others from land o Self-help concerns – worried about people taking the law into their own hands to retaliate •Reinforces absolute right to excludeWhat is at stake behind the nearly absolute right to exclude?•Right to exclude protects right to economically productive useo Economic development →people probably wouldn’t invest in their land if someone else could come in and damage it•Property right also protect privacy, autonomy, liberty interestso Invasion of property regarded as an invasion of personal space – personal wrongo Historical civic value → land ownership meant citizenship, right to vote•Safety and security concerns → controlling your physical space gives you extra physical safety and security → If someone is willing to pay the price of trespass, they can still get what they want—but the price is very high When do LOs lack the right to exclude? (absolute right to exclude not so absolute after all) •Eminent domain – the state can trump everything•Easements –right to cross someone else’s propertyo i.e. if there is an established shortcut, right of way•Necessity – if life or limb is at stake, may have a privilege of necessity•State and its agents can enter private property for public purposes•Discrimination / public accommodations –can’t exclude on prohibited grounds if you are admitting any members of the public onto land•Free speech rights on private propertyo i.e. handing out pamphlets sometimes allowed constitutionally→ If you invite others onto your property to make use of it, you are subject to many more exceptionsHinman v. Pacific Air Transport (9th Cir, 1936)•Issue: rights of LO in connection w/ the flight of aircraft above his lando Ad coelum→“whoever owns the soil owns also to the sky and to the depths”▪So could be a trespass b/c of frequent invasion by planeso LOs claim they have 150 ft under the ad coelum rule, b/c that is what they could make use of •Holding: “owner of land owns as much of the space above him as he uses, but only so long as he uses it”o Don’t own the air space unless you have exercised dominion over it▪Don’t need to have actual dominion over surface land – you are deemed to be inconstructive possession of ito To exercise dominion → can build up, or engage in activities on the surface that use the air space ▪Would have nuisance claim if there was some loss of enjoyment of land b/c of low flights •Superjacent air space not protected w/ an absolute right of exclusiono Need to show dominion over air space or serious impairment of activity / use of lando Court rejects ad coelum rule → says it came about before plane travel, and would result in an absurdity to regulate who owns air spaceUS v. Causby (1942) – recognizes fed gov control over navigable airspace•Airspace in which planes travel is a type of public property, in which no surface O has any claim of private property rights•This is not a taking unless the flights are so low as to destroy the use and enjoyment of the surface areaNuisanceA non-trespassory invasion or interference in another’s use and enjoyment of land•May be negligent, reckless, abnormally dangerous, or intentional + unreasonable•Unreasonable = gravity of the harm outweighs utility of the activity causing harmo Restatement (2nd) of Torts § 827 “gravity of harm” factors →▪(a) extent of the harm▪(b) character of the harm▪(c) social value that the law attaches to the type of use or enjoyment invaded▪(d) suitability of the particular use and enjoyment invaded to the character of the locality▪(e) burden on the person harmed of avoiding the harmo Restatement (2nd) of Torts § 828 “utility of conduct” factors →▪(a) social value that the law attaches to the primary purpose of the conduct▪(b) suitability of the conduct to the character of the locality▪(c) impracticability of preventing or avoiding the invasionHendricks v. Stalnaker (WV, 1989)•Issue: Stalnaker’s well created a buffer zone of 100 feet –Hendricks didn’t have any other options for their placement of their septic systemo Stalnaker argues that b/c well is not an unreasonable use of his land, he is not liable for effects on Hendricks’ property•Holding: water well is non an unreasonable use of land, and therefore is not a private nuisance o Balancing of LO’s interests is at least equa l, or perhaps slightly in favor of the well▪Septic tank is a little more invasive – seeps out liquid, rather than drawing water in▪Invasion is a part of the definition of nuisance doctrineConceptions of Property – Philosophical Perspectives [p.I-15]•Essentialists (Blackstone) → seek to uncover the single true definition of property as a legal concept o i.e. Jacque– broad rights afforded to LOs to open their gates as they see fit •Skeptics → think of property as a bundle of sticks (rights), meaningless until spelled out by legal system o i.e. Hendricks – rights are fact-specific and gov may interfere to determine outcome •Hinman falls in the middle – pragmatic approach to propertyWhy do we protect the right to exclude w/ a nearly absolute rule, and the right to enjoy w/ a squishy balancing test?•We grant absolute right to be free from physical invasions –but can’t grant such a right re: non-physical invasions b/c neighbors are entitled to use their own propertyo So nuisance law is more fact-specific•Granting absolute right to be free from nuisance invasions would stop a lot of socially productive activityExclusion and GovernanceCan view trespass and nuisance as 2 different strategies for resolving disputes about how scarce resources are used •Exclusion→ decisions about resource use are delegated to an O who acts as gatekeeper of the resource o Likely to be favored where particular resources like land have multiple potential uses, and when we think it desirable to give Os discretion to choose which use is most valuable▪i.e. Jacqueo Judges uphold owners’ rights•Governance→ focuses on particular uses of resources, and prescribes particular rules about permitted and prohibited uses without regard to the other attributes of the resourceo Tends to be used in situations where the particular uses of property are of heightened significance, either b/c they are strongly favored or disfavored▪i.e. Hendrickso Judges determine how property will be usedThe Coase Theorem•Reciprocal harm → should A be allowed to harm B, or should B be allowed to harm A?o The problem is to avoid the more serious harm•Jacque → ex ante the Jacques could have consented and charged anything less than what it would have cost Steenberg to take the private road (b/c it wouldn’t have caused them substanti al harm)o Why didn’t they?▪Their subjective valuation may have been greater than what Steenberg was willing tooffer, or there may be have been bad blood b/w parties that hindered bargaining o Not the rational maximizer from Coase’s Theorem – so it matters where the law places the entitlement, even if parties could have bargained differentlyPROPERTY & EQUITYRepeated Trespasses →Baker v. Howard County Hunt (MD, 1936)•Issue: can plaintiff get injunction against repeated trespasses which, while not continuous, are part of a single course of conduct that seriously interferes w/ O’s right to the peaceful enjoyment of his property?•Holding: injunction justified against continuing or repeated trespasses in pursuance of a single purpose o Case meets exceptions to ru le that equity won’t enjoin a mere trespass →▪Where the injury is irreparable, or where full and adequate relief can’t be granted at law, or to prevent repetition of similar action / multiplicity of suitso Here can’t compensate for the injuries done – inter ruption of Mr. Baker’s rabbit experiment and the farm as a peaceful refugeo Also def manifested an intention to persist in trespassing – would be costly and time-consuming to prosecute each act of trespass•Also general rule that entry by dogs on another’s l and = trespass only if you have reason to believe that they might do damage, or if your voluntary actions cause / facilitate the dogs being thereo Hunting is a valuable, useful activity –would constrain it too much if we didn’t allow dogs to inadvertently cross another’s land▪Resembles Hinman treatment of air travel – have to make exception to the absolute right to exclude lest we shut down the activity entirelyCould Baker court have granted punitive damages (like Jacque court)?•May deter Hunt Club from g oing through Baker’s land without shutting it down•But might not be effective if Club discounted the possibility as being improbable and continued trespassing o Also punitive damages are useless if Club is judgment-proof (has no $ / assets)Property Rules vs. Liability Rules•Manner in which entitlements are protected –Calabresi & Melamed [p.I-54]o Property rule –right can’t be taken away unless you give consent (except thru eminent domain)▪Involves decision as to who gets initial entitlement – but not as to its value o Liability rule – right can be taken away at a price set by the court▪Not only are entitlements protected, but their transfer / destruction allowed on the basis ofa value determined by the state rather than by the parties themselveso Inalienablity rule – transfer not permitted b/w willing buyer and willing seller▪Not only protects entitlement, but also may be viewed as limiting or regulating the grant of the entitlement itself•Courts have a choice along 2 dimensions →o(1) Who gets entitlement? – plaintiff or defo(2) How is it protected? – property rule or liability ruleBuilding Encroachments →Pile v. Pedrick (PA, 1895)•Issue: defs mistakenly build wall foundation 1 3/8” onto plaintiff’s property (below the surface) o Plaintiffs refused to grant permission to defs to come onto their property to chip away intrusion •Holding: Rule 1 – plaintiffs granted absolute property right to consent or not, or set their own price in bargaining (like Jacques)o Court gave plaintiff option to get injunction (building must be torn down) or get damages▪Doesn’t matter that intrusion was unintentionalo Plaintiff chooses injunction – may want to prevent defs from operating the factory altogether ▪Nuisance claim would be hard to make out b/c of utility of factor▪Also injunction may be the starting point for bargaining – could get any value b/w harm done to them (small amount) and cost of taking down the building (large amount) Golden Press v. Rylands (CO, 1951)•Issue: plaintiffs contend that foundation of def’s building ext ends 2–3 ½” onto their lando Ask court to issue injunction requiring that building be torn down•Holding: Rule 2 – plaintiff has entitlement but compensated by compensatory damages to the extent of their injuries (no injunction)o“Where the encroachment is deliberate and constitutes a willful and intentional taking of another’s land, equity may well require its restoration regardless of the expense of removal as comparedwith damage suffered therefrom; but where the encroachment was in good faith, we think the courtshould weigh the circumstances so that it shall not act oppressively.”▪No universal rule as to injunctions for encroachment – sometimes compensatory damages▪Mandatory injunction may be properly denied when def’s encroachment is unintentional and sli ght, plaintiff’s use is not affected and his damage small and fairly compensable,while cost of removal is so great as to cause hardship or make removal unconscionable Which rule is best for minimizing trespassing while bearing cost-benefit in mind –Pile or Golden Press?•Pile induces extra investment in precautions – puts all the onus on the person buildingo Creates incentive for neighbor to maximize bargaining power and wait until most of the building is up to raise trespass claim•Golden Press supports equity rule against injunction if person seeking it has unclean handso Under doctrine of laches, if you wait too long and the party invading your rights relies on your not enforcing them, you can’t get an injunction•In some circumstances, may make sense to split the entitlement – hybrid property-liability rule o Make def and plaintiff share the cost of taking down the wall to encourage honestyCoasean analysis•Pile → if cost of removal greatly exceeds harm from encroachment, parties will bargain so entitlement goes to def, and the wall stays upo Efficient result would be for def to offer compensation up to the amount of tearing down the wallo Probably won’t happen in this case b/c plaintiffs don’t want structure at all•Golden Press → plaintiff has to sell entitlement for a trivial amount to money set by the court o If having wall taken down worth more to plaintiffs than having it for defs, ideally would pay enough to get it taken downo Might not happen if defs don’t have enough moneyLikely in both cases that the entitlement will stay where the court assigned it – so should think about where they want it to end upHow does one acquire property (other than by gift or purchase)?BY CAPTURE / OCCUPANCY (First Possession)First possession is based on being the first to possess an unclaimed thingWild Animals → previously un-owned propertyPierson v. Post (NY, 1805)•Issue: Post says Pierson took his fox – argues he had first possession of it•Holding: first possession not established – has to show occupancy over the fox to get titleo Occupancy requires possession / capture → meaning you have deprived the animal of its natural liberty and brought it within your certain control in tangible ways▪Rule of capture –capture or “certain control”▪Property in animals ferae naturae may be acquired provided the pursuer be within reach or have a reasonable prospect or taking the animal (i.e. mortal wounding orcapturing within nets, traps, etc)o Dispute wasn’t really about the fox, but the value of the hunt – Post thought he was legally entitled to the fox b/c the custom of hunting said he was entitled to finish hunting it •Livingston’s dissent – says the facts have amounted to Post having pursuit of a reasonable prospect of capture, and that is sufficient (maj would say reasonable prospect of capture not sufficient)o Chooses the rule that will better encourage fox hunting (calls foxes enemies of the human race) ▪Who would invest in hunting foxes if, at such a late point in the hunt, another huntercould swoop in and claim it?▪But may not be the best policy –can’t always know when you start a hunt whethersomeone is already hunting that fox•Policy justification for rule of capture → designed to minimize disputes b/c unlikely that more than one person can claim such possession / captureWhat if Pierson and Post were trespassing on another person’s land?•Competing claims of first possession (fox hunter) and ratione soli (LO)•Landowners are deemed to have constructive possession of wild animals who are on their property, as against a trespasser – even if that trespasser captured the animalo Because of importance of discouraging trespass to land▪Justifies punitive damage award in Jacque, and overrides rule of capture •Why not say Pierson would get the fox, and LO could sue him for trespass?o Don’t want people to trespass if they deem that it’s worth it – i.e. if only had to pay negligible damages, may run onto another’s land if chasing something very valuableKeeble v. Hickeringill (Eng, 1809)•Issue: def shot bullets into plaintiff’s duck pond, whic h was plaintiff’s source of livelihood•Holding: even if you can show malicious intent, must also show that such action wasn’t done in pursuit of one’s own livelihood – must show malicious intent that is bad for society, not one thatmerely creates fair competitiono Early example of distinction b/w actionable interference of business vs. unfair competition (malicious interference w/ economic activity)▪Interference w/ business activity became the basis for asserting property rights – courthad to carve out an exception for fair competition•Re-argument of Pierson v. Post in light of Keeble?o Post could say he was engaging in a lawful and useful pursuit as part of his livelihood, and that Pierson interfered solely to avoid Post’s capture of the fox▪But Pierson could say he was competing for the fox in pursuit of his livelihood, whichwould mean he would still win under Keebleo Have to decide if Post is an example of unfair competition or not▪Just desert reason – did Post earn protection from this kind of reason?▪Instrumental reason – does society benefit from this kind of competition?o Keeble would have been a better argument –probably would have helped Post’s sideGhen v. Rich (MA, 1881)•Issue: whaler killed whale; it was marked when finder found it, but finder kept it•Holding: whaler granted possession of whaleo Whaling custom establishes that the whale killer owns it, not the person who finds it – whaler shoots whale and marks it, pays finder’s fee to person who finds ito Court says unless this is sustained, whaling would cease b/c no one would engage in it if the fruits of his labor could be appropriated by any chance finder (like Livingston’s dissent in Pierson) ▪But how can court justify this outcome in light of the Pierson rule of capture?•Translate custom into what should count as capture given the nature of the thingat stake and the industry’s dependence on this custom•If arguing for whalers under Pierson v. Post theory → they mortally wounded the whale w/ intent to come back and get it – does this count as bringing it within their certain control?o Economic policy considerations: concerned about rewarding and encouraging productivity, and concerned about free riderso Also concerned about minimizing disputes, maintaining social orderCapture = what counts as having it first•May want to encourage the investments that go into captureo Ghen, Keeble, Pierson dissent•Or may want to encourage the race to captureo Pierson majorityPrinciples that the courts rely on in dealing w/ resource allocation of previously un-owned property •Title is relative → who had better claim as between the parties?o Who had something like possession first in time? (legal conclusion rather than a brute fact) •Historical dimension → finding principles that are out there already and applying them to particular case o Pierson v. Post majority: law is out there to be applied (Blackstonian view)▪CL reflects accumulating but unchanging wisdom•But if follow universal rule (Pierson), create a world of freeloaders b/creturn of investment goes to someone who invested nothingo Pierson v. Post dissent: should adopt a rule that serves our interests here and now▪CL as an instrument for the governance of society – should accommodate the changefrom a rural to an industrial society•If use customary rule (Ghen), return allocated w/ respect to cost / riskassociated w/ the investment → universal contract by way of custom •Societal concerns re: allocation of resources dictate decisions, but thru the language of prior decisions o Court invents constructive possession to allow for first possession rule to prevail, whileacknowledging the fundamental property right re: landEpstein → how do we sort out the conflict b/w Pierson and Post?•Trying to get the right level of investment by giving property right to the person who hunts first o Front-end investment considerations > than after-the-fact property determinationso He is inclined toward the dissent•Micro issue → which optimal rule will best assign property rights?o Choosing the right rule will lead to the faster destruction of the animal species, so see leg rules in response to 1st possession rules – designed to maximize hunting while recognizing concerns ofendangered species, etc.Open Access and the Commons– Eggertsson article•Open access = access is open to all members of a particular community, but not to all outsiders o Rule of capture immediately establishes ownership of a resource once it is withdrawn (i.e. fisher owns fish he catches in an open access fishery)•Open access problems arise when independent actors have the incentive and ability to withdraw resource unites from an asset they access togethero Then all will act as free riders – depleting from investment rather than trying to improve ito Actors will only think of their slight gain, not the net loss to society•Internal arrangements or control by a private owner / the state necessary to regulate open access resources Commons, Anticommons, and Semicommons•Open access regime →“tragedy of the commons” b/c no one has the right to exclude anyone else, and thus everyone will exercise his right of access•Anticommons →“tragedy of the anticommons” b/c too many have the right to exclude, and thus no one is able to use the resourceo But giving people a veto over the development of a resource (i.e. a park) can help secure the commitment to preserve the park for future generations•Semicommons → occurs when a given resource is subject to private exclusion rights in some uses, but is open access for other purposeso But people will have the incentive to favor their part of the semicommons and trash the other parts Other Applications of First Possession → abandoned or lost propertyEads v. Brazelton (Ark, 1861)•Issue: Brazelton found sunken ship –marked it but couldn’t raise it then b/c of high tides, so left w/ intent to return for it; Eads found ship 10 months later and raised it himselfo Brazelton says right of occupancy vested in him by his discovery of the wreck and his intentions •Holding: Brazelton never took possession of the wreck – since he left the site, he no longer has a claim to it o His intention to possess deemed useless without detention of the property▪Need actual taking + intent to possess → certain degree of due diligence required in first possession claims•Case decided in light of Pierson v. Post – firmer possession needed o Adjudged abandonment based on Brazelton’s actions and the length of time passed▪Epstein: get rid of this abandonment stuff – should use salving rules and create a statute of limitationsPopov v. Hayashi (CA, 2002)•Issue: argument over who has possession of a very valuable home-run baseballo Once baseballs leave playing field they are regarded as being abandoned property – can be claimed by the first person to take possession of them•Holding: ball should be sold, with ½ proceeds going to Popov and ½ to Hayashi (this is unprecedented –common law hates shared possession unless it is the product of prior agreement)o Says that although Popov can’t prove that he established possession, once it landed in his mitt he had an exclusive “pre-possessory” interes t in being allowed to complete catch without interference ▪The mob interfered w/ this interest – like the malicious interferers in Ghen and Keeble▪But Hayashi wasn’t part of the mob / malicious interferor, and was the first tounambiguously establish possession of the ballo“Both men have a superior claim to the ball as against all the world. Each man has a claim of equal dignity as to the other.”▪Rule gives both parties first possession rights –but then doesn’t let either keep it •Estlund: judge should have ruled for Hayashi –Popov couldn’t bear the burden of proof•Epstein: judge should have based his ruling on Ghen and said Popov had done everything he could under the circs to establish possessiono Shouldn’t matter that Hayashi didn’t actually interfere w/ Popov – rule is whether he knew that Popov already had a claim to the ball, which he clearly didHammonds v. Central Kentucky Natural Gas Co. (KY, 1934)•Issue: who gets to take oil which is located in common under multiple people’s land?•Holding: if gas “turned loose in the earth and wandered into plaintiff’s land,” def is not liable for the value of the use of her property, b/c ceased to be the exclusive owner of the gas – it became mineral farae naturae o Rule of capture is CL rule for oil and gas → anything you can get from extracting under your own land is yours▪Unless can show slant drilling –which is a trespass b/c drills oil from other’s land o But this case involved oil re-injected into land → is this a trespass?▪Court says no – re-injecting releases it into the wild, returns it to its natural state, sosurface O can extract it and use it•Analogizes re-injected oil to wild animals▪Oil and gas belong to O for use and possession – but belong to no one for purposes ofbeing underground / trespassPossession depends on the nature of the thing possessed, and the industry involved•Law looks at →o Who made most significant investment into getting the thing into society?o Who is acting as a free rider?•“such possession as the nature of the thing / its situation allowed or permitted under circs”BY CREATIONCreation involves being the first to possess some new or novel thingPersons who create new information are (sometimes) rewarded w/ a right to exclude others from using that info •The primary reason for creating property rights in info is to provide incentives for producing more of it –not to assure that it is allocated efficiently among potential users of ito Need to protect so there is incentive to create, but not overprotect which would limit competition and raise priceso CL basis and statutory regimes▪Limited monopoly rights granted by legislature (enumerated powers under Art I) •Temporal element – being first is important•Some type of originality / novelty also required for protectiono Though often difficult to distinguish the novel from merely being the first of its kind Misappropriation and the Quasi-Property Right in Hot News →INS v. AP (US, 1918)•Issue: INS took reports published by AP and reprinted them as their owno AP wants to enjoin as a taking of their property – says misappropriation of the quasi-property made the competition unfair▪Quasi-property claim → have right to stop competitors from taking “fresh” newsreports and republishing as their own (rt against specific parties—not the whole world) o INS claims AP abandoned their property by publishing it – so it was common property •Holding: court finds quasi-property right / unfair competitiono Says this is unauthorized interference w/ AP’s legitimate business precisely at the point where profit is to be reaped in order to divert part of the profit to themselves▪Says AP abandoned news to the public – but not to competitors using it for themselves▪The gains / profits are at issue, not the property itself•Brandeis’ dissent – says to appropriate and use for profit knowledge and ideas produced by another, without making compensation or acknowledgement, may be inconsistent with a sense of propriety – butthat the law has heretofore sanctioned the practice (w/ some exceptions)o Previous unfair competition cases have relied on fraud or malicious intento Concerned that maj’s ruling will limit many readers’ access to the newsNBA v. Motorola (2nd Cir, 1997)•Issue: deals with copyright claim – copyright law preempts state claims based on misappropriation of underlyi ng games, but narrow “hot news” exception from INS. AP survives preemptiono NBA says Motorola misappropriated “hot news”o Motorola must distinguish it’s conduct from INS’ conduct•Holding: Motorola’s transmission of “real-time” NBA game scores / info does no t constitute a misappropriation of “hot news” that is NBA’s property。

高级英语第三版第一册课后英译汉答案解析

高级英语第三版第一册课后英译汉答案Unit1Paraphrase:1. We’re 23 feet above sea level.2. The house has been here since 1915, andno hurricane has ever caused any damage to it .3. We can make the necessary preparations and survive the hurricane without much dam age.4. Water got into the generator and put it out. It stopped producing electricity, so the ligh ts also went out.5. Everybody goes out through the back door and runs to the cars!6. The electrical systems in the car (the battery for the starter) had been put out by water.7. As John watched the water inch its way up the steps, he felt a strong sense of guilt bec ause he blamed himself for endangering the whole family by deciding not to flee inland.8. Oh God, please help us to get through this storm safely9. Grandmother Koshak sang a few words alone and then her voice gradually grew dimm er and finally stopped.10. Janis displayed the fear caused by the hurricanerather late.1. 每架飞机起飞之前必须经过严格的检查。

On Entropy and Monotonicity for Real Cubic Maps

On Entropy and Monotonicity for Real Cubic Maps John Milnor and Charles Tresser1 with an appendix by Adrien Douady and Pier9096v2 [math.DS] 25 Jun 1999
On the other hand, it has been proved more recently that any value of the topological entropy in [0, log 2] which cannot be achieved by a map having a periodic critical orbit, is realized for a single value of v . ([Gs], [L2].) This result is a special case of the Generic Hyperbolicity Conjecture, which states that every rational map can be approximated arbitrarily closely by a rational map (real if the original map is real, and a polynomial if the original is a polynomial) such that the orbits of all critical points converge to attracting periodic orbits. (For an early version on this conjecture, see Fatou [F].)
Contents 1. Introduction 2. Piecewise monotone maps and kneading theory 3. Parametrization of polynomial maps 4. Topological entropy and periodic orbits 5. The stunted sawtooth family 6. Contractibility of isentropes for the stunted sawtooth family 7. Bones in P 2 8. Monotonicity, intersections of bones, and the period n -skeleton 9. From connected bones to connected isentropes Appendix A. (by A. Douady and P. Sentenac): Characterization of a polynomial by its critical values Appendix B: Tight symbol sequences and Thurston’s theorem Appendix C. Monotonicity versus antimonotonicity References §1. Introduction Consider a continuous map f from a closed interval I to itself. In the simplest cases, I is the union (in a not necessarily unique way) of sub-intervals where f is monotone. The minimal number of sub-intervals required (called the lap number ) is a rough measure of the complexity of the map f . If now we think of f as generating a dynamical system, a quantitative description of the dynamic complexity of f is obtained by measuring the rate of exponential growth of the lap numbers of the successive iterates f ◦1 = f , f ◦2 = f ◦f , . . . of the map f . This growth rate is clearly invariant under a continuous change of coordinate, and turns out to be equal to a measure of dynamic complexity used in a much more general context and known as the topological entropy h(f ) . Maps on the interval provide the simplest examples to illustrate the problem of understanding how dynamic complexity evolves under deformations of a dynamical system. However, even the basic problem of comparing the topological entropies of two smooth interval maps which are close to each other is only partly understood. The best results so far

1.1 On The Promise of Bayesian Inference for

tatistics in Astronomical Investigation
3
admitting uncertainty in terms of a prior distribution for (bjI). Also, expecting low counts implies b will be rather small (see also the paper by John Nousek, this volume, in discussion of low count radiation from SN1987A). Using a prior p(sjbI) that is uniform over a `large' range (and does not depend on b), Loredo proceeds to summary inferences based on the posterior p(sjnbI) in his equation (5.13). Throughout the paper, such uniform priors are adopted as a routine on the basis of representing suitable forms of `ignorance' about the quantity concerned. If any area of Bayesian inference has received too much attention during the last couple of decades it is surely the search for unique and `objective' representation of ignorance { see 4] for a recent and partial review of the eld. The maximum entropy school has been in uential in physical sciences, as referenced by Loredo, and particularly predominant in expounding the view that a single prior may be found, in any given situation, to represent vagueness in the sense of maximum entropy subject to certain `plausible' assumptions that typically stand for little more than mathematical convenience in determining a unique solution in the resulting MaxEnt framework. There is nothing unique, objective or otherwise scienti cally persuasive about uniform priors for location parameters, or any of the plethora of vague, reference or indi erence priors that abound. In investigations which admit an `objective' (de ned simply as consensus of informed observers) data model as here (ie. p(njsI)), analysis should necessarily involve study of sensitivity to qualitative and quantitative aspects of the prior, including assessments of pre-data predictive validity of the fdata modelg:fpriorg combination, and post-data determination of the mapping from prior to posterior for ranges of scienti cally plausible priors. The issue of pre-data validity is addressed through the implied (prior) R predictive distribution for the data, here p(njbI) = p(njsI)p(sjbI)ds. When n is observed, the value of this density function provides the normalising constant in Bayes' theorem (C ?1 in Loredo's equation (5.6)). Prior to the data, however, this distribution describes the investigator's view of experimental outcome. A uniform prior over a very large range translates essentially into a similar (though discrete) uniform p(njbI), which most observers would be quite concerned about as a plausible and scienti cally valid representation of expectations. The issue is particularly acute in problems of low counts and source detection when s (when non-zero) will be tend to be small | reasonable priors for s, and thus predictions about n, should surely re ect this. Competing `reference' priors (and there are many { 4]), lead to posteriors that can di er markedly with low counts s, though all such priors claim some form of `vagueness' and `uniformity' (on some scale). Scienti c investigation must involve careful and thorough consideration of initial information, modes of incorporation of such information in summary inferences, and exploration of sensitivity to prior assumptions (which includes model and data assumptions and well as priors for model parameters { and sometimes the distinction is unclear and even irrelevant

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† ∗
proof. Clearly, if we do not give a complete proof to a veriﬁer (that does not have the power or time to generate and check the proof), then we cannot expect the veriﬁer to be completely convinced that the theorem is provable. This led to a very fascinating problem: how can a veriﬁer be convinced with high probability that a given theorem is provable without seeing the whole proof? and how rapidly can this be done? This problem has been formulated and extensively studied in terms of interactive protocols [Gol89]. Informally, an interactive protocol consists of a Prover and a Veriﬁer. The Prover is an all powerful Turing Machine (TM) and the Veriﬁer is a TM which operates in time polynomial in the length of the input. In addition, the Veriﬁer has a random source (e.g., a fair coin) not visible to the Prover. In the beginning of the interactive protocol the Prover and the Veriﬁer receive the same input string. Then, the Prover tries to convince the Veriﬁer, through a series of queries and answers, that the input string belongs to a given language. The Prover succeeds if the Veriﬁer accepts with probability greater than 2/3. The probability is computed over all possible coin tosses made by the Veriﬁer. However, the Veriﬁer must guard against imposters masquerading as the real Prover. That is, the Veriﬁer must not be convinced to accept a string not in the language with probability greater than 1/3—even if the Prover lies. Deﬁnition Let V be a probabilistic polynomial time TM and let P be an arbitrary TM. P and V share the same input tape and communicate via a communication tape. P and V form an interactive protocol for a language L if 1. x ∈ L =⇒ Prob[ P -V accepts x ] > 2/3. 2. x ∈ L =⇒ ∀P ∗ , Prob[ P ∗ -V accepts x ] < 1/3. A language L is in IP if there exist P and V which form an interactive protocol for L. Clearly, IP contains all NP languages, because in polynomial time the Prover can give the Veriﬁer the entire proof. In such a protocol, the Veriﬁer cannot be fooled and never accepts a string not in the language. To illustrate how randomness can generalize the concept of a proof, we look at an interactive protocol for a language not known to be in NP. Consider GNI, the set of pairs of graphs that are not isomorphic. GNI is known to be in co-NP and believed not to be in NP. However, GNI does have an interactive protocol [GMW86]. For small graphs, the Veriﬁer can easily determine if the two graphs are not isomorphic. For suﬃciently large graphs, the Veriﬁer solicits help from the Prover to show that Gi and Gj are not isomorphic, as follows: 1. The Veriﬁer randomly selects Gi or Gj and a random permutation of the selected graph. This process is independently repeated n times, where n is the number of vertices in Gj . If the graphs do not have the same number of vertices, they are clearly not isomorphic. This sequence of n randomly chosen, randomly permuted graphs is sent to the Prover. Recall that the Prover has not seen
This research was supported in part by NSF Research Grant CCR 88-23053. Supported in part by an IBM Graduate Fellowship. Current Address: Department of Computer Science, University of Maryland, Baltimore County Campus, Baltimore, MD 21228, USA. ‡ Current Address: Max Plank Institut f¨ ur Informatik, Im Stadtwald, W 6600 Saarbr¨ ucken, Germany. On leave from Department of Computer Science, New Mexico ቤተ መጻሕፍቲ ባይዱtate University, Las Cruces, NM 88003, USA.
the Veriﬁer’s random bits. This assumption is not necessary, but simpliﬁes the exposition. 2. The Veriﬁer asks the Prover to determine, for each graph in the sequence, which graph, Gi or Gj , was the one selected. If the Prover answers all the queries correctly, then the Veriﬁer accepts. Suppose the two original graphs are not isomorphic. Then, only one of the original graphs is isomorphic to the permuted graph. The Prover simply answers by picking that graph. If the graphs are isomorphic, then the Prover has at best a 2−n chance of answering all n questions correctly. Thus, the Veriﬁer cannot be fooled with high probability. Therefore, GNI ∈ IP. Note that GNI is believed to be incomplete for co-NP. So, the preceding discussion does not show that co-NP ⊆ IP. For a while, it was believed that co-NP is not contained in IP, because there are oracle worlds where co-NP ⊆ IP [FS88]. In fact, the computational power of interactive protocols was not fully appreciated until Lund, Fortnow, Karloﬀ and Nisan [LFKN90] showed that IP actually contains the entire Polynomial Hierarchy. This result then led Shamir [Sha90] to completely characterize IP by showing that IP = PSPACE. Babai, Fortnow and Lund [BFL90] characterized the computational power of multiprover interactive protocols MIP = NEXP. In both cases, it is interesting to see that interactive proof systems provide alternative deﬁnitions of classic complexity classes. Thus, they ﬁt very nicely in the overall classiﬁcation of feasible computations. Furthermore, both of these problems have contradictory relativizations [FS88]. That is, there exist oracles A and B such that IPA = PSPACEA and IPB = PSPACEB , and similarly for the multi-prover case. Thus, these results provide the ﬁrst natural counterexamples to the belief that problems with contradictory relativizations are beyond our proof techniques. The IP = PSPACE result also provides a very dramatic counterexample to the already battered Random Oracle Hypothesis. In [HCRR90], we showed that ProbA [ IPA = PSPACEA ] = 1.