DOCSLIB.ORG

Groups of Piecewise Projective Homeomorphisms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Groups of Piecewise Projective Homeomorphisms Groups of piecewise projective homeomorphisms Nicolas Monod1 Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Edited by Gregory A. Margulis, Yale University, New Haven, CT, and approved February 5, 2013 (received for review October 22, 2012) The group of piecewise projective homeomorphisms of the line The main result of this article is the following, for which we provides straightforward torsion-free counterexamples to the so- introduce a method for proving amenability. called von Neumann conjecture. The examples are so simple that many additional properties can be established. Theorem 1. The group H(A) is nonamenable if A ≠ Z. The next result is a sequacious generalization of the corre- free groups | paradoxical decomposition | von Neumann problem sponding theorem of Brin–Squier (16) about piecewise affine transformations, and we claim no originality. n 1924, Banach and Tarski (1) accomplished a rather para- Idoxical feat. They proved that a solid ball can be decomposed Theorem 2. The group H does not contain any nonabelian free sub- fi group. Thus, H(A) inherits this property for any subring A < R. into ve pieces, which are then moved around and reassembled = R in such a way as to obtain two balls identical to the original one Thus, already H H( ) itself is a counterexample to the von Neumann conjecture. Writing H(A) as the directed union of its (1). This wellnigh miraculous duplication was based on Haus- fi dorff’s (2) 1914 work. nitely generated subgroups, we deduce Corollary 3. In his 1929 study of the Hausdorff–Banach–Tarski paradox, Corollary 3. For A ≠ Z, the groups H(A) contain finitely generated von Neumann (3) introduced the concept of amenable groups. subgroups that are simultaneously nonamenable and without non- Tarski (4, 5) readily proved that amenability is the only ob- abelian free subgroups. struction to paradoxical decompositions. However, the known paradoxes relied more prosaically on the existence of nonabelian Additional Properties. The groups H(A) seem to enjoy a number free subgroups. Therefore, the main open problem in the subject of additional interesting properties, some of which are weaker remained for half a century to find nonamenable groups without forms of amenability. In the last section, we shall prove the fol- free subgroups. Von Neumann’s (3) name was apparently at- lowing five propositions (and recall the terminology). Here, A < R tached to it by Day in the 1950s. The problem was finally solved is an arbitrary subring. around 1980: Ol′shanskiĭ (6–8) proved the nonamenability of the Tarski monsters that he had constructed, and Adyan (9, 10) Proposition 4. All L2-Betti numbers of H(A) and G(A) vanish. showed that his work on Burnside groups yields nonamenability. Finitely presented examples were constructed another 20 y later Proposition 5. The group H(A) is inner amenable. by Ol′shanskiĭ–Sapir (11). There are several more recent coun- terexamples (12–14). Proposition 6. The group H is biorderable, and hence, so are all of its Given any subring A < R,weshalldefine a group G(A)anda subgroups. It follows that there is no nontrivial homomorphism subgroup H(A) < G(A) of piecewise projective transformations. from any Kazhdan group to H. Those groups will provide concrete, uncomplicated examples with many additional properties. Perhaps ironically, our short Proposition 7. Let E ⊆ P1 be any subset. Then, the subgroup of proof of nonamenability ultimately relies on basic free groups H(A), which fixes E pointwise, is coamenable in H(A) unless E is of matrices, as in Hausdorff’s (2) 1914 paradox, although the dense (in which case, the subgroup is trivial). Tits (15) alternative shows that the examples cannot be linear themselves. Proposition 8. If H(A) acts by isometries on any proper CAT(0) space, then either it fixesapointatinfinity or it preserves a Euclidean subspace. Construction One can also check that H(A) satisfies no group law and has vanishing properties in bounded cohomology (see below). I saw the pale student of unhallowed arts kneeling beside the thing he Nonamenability had put together. An obvious difference between the actions of PSL2(A) and H(A) on P1 is that the latter group fixes ∞, whereas the former does Mary Shelley, introduction to the 1831 edition of Frankenstein not. The next proposition shows that this difference is the only Consider the natural action of the group PSL2(R)onthe one as far as the orbit structure is concerned. projective line P1 = P1(R). We endow P1 with its R-topology, 1 making it a topological circle. We denote by G the group of all Proposition 9. Let A < R be any subring, and let p ∈ P \{∞}. Then, 1 homeomorphisms of P that are piecewise in PSL2(R), each piece 1 · ⊆ ∞ ∪ · : being an interval of P with finitely many pieces. We let H < G PSL2ðAÞ p f g HðAÞ p be the subgroup fixing the point ∞∈P1 corresponding to the first basis vector of R2. Thus, H is left-orderable, because it acts Thus, the equivalence relations induced by the actions of PSL2(A) P1 P1 ∞ faithfully on the topological line P1\{∞}, preserving orientations. and H(A) on coincide when restricted to \{ }. It follows in particular that H is torsion-free. 1 Given a subring A < R, we denote by PA ⊆ P the collection of Author contributions: N.M. designed research, performed research, and wrote the paper. all fixed points of all hyperbolic elements of PSL2(A). This set is The author declares no conflict of interest. PSL2(A)-invariant and countable if A is so. We define G(A)tobe the subgroup of G given by all elements that are piecewise in This article is a PNAS Direct Submission. PSL2(A) with all interval endpoints in PA. We write H(A) = G(A) Freely available online through the PNAS open access option. ∩ H, which is the stabilizer of ∞ in G(A). 1E-mail: nicolas.monod@epfl.ch. 4524–4527 | PNAS | March 19, 2013 | vol. 110 | no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1218426110 Downloaded by guest on October 1, 2021 Proof. We need to show that, given g ∈ PSL2(A)withgp ≠∞, nonamenable (see also Remark 10 and Remark 11). Equiva- ∈ = 1 there is an element h H(A), such that hp gp. We assume lently, the Γ-actiononPSL2(R) is nonamenable. Viewing P as ∞≠∞ = g , because otherwise, h g will do. Equivalently, we a homogeneous space of PSL2(R), it follows that the Γ-action on need an element q ∈ G(A) fixing gp andsuchthatq∞≠g∞, P1 is nonamenable. Indeed, amenability is preserved under −1 writing h ≠ q g.Itsuffices to find a hyperbolic element q0 ∈ extensions (ref. 21, Corollary C or ref. 23, 2.4). This action is a. 1 PSL2(A)withq0∞ = g∞ and fixed points ξ± ∈ P that separate e.-free, because any nontrivial element has, at most, two fixed gp from both ∞ and g∞ (Fig. 1). Indeed, we can then define q points. Thus, the relation induced by Γ on P1 is nonamenable. 1 1 to be the identity on the component of P \{ξ±} containing gp Restricting to P \{∞}, we deduce from Proposition 9 that the fi and de ne qto coincide with q0 on the other component. relation induced by the H(A) action is also nonamenable. ab Let be a matrix representative of g; thus, a, b, c, d, ∈ A, [Amenability is preserved under restriction (ref. 18, 9.3), but here, cd {∞} is a null-set anyway.] Thus, H(A) is a nonamenable group.□ − = ∞≠∞ ≠ and ad bc 1. The assumption g implies c 0, and ab+ ra Remark 10. We recall from ref.22that the nonamenability of the thus, we can assume c > 0. Let q0 be given by with cd+ rc Γ-relation on PSL2(R) is a general consequence of the existence of Γ r ∈ A to be determined later; thus, q0∞≠g∞. This matrix is a nondiscrete nonabelian free subgroup of . Thus, the main point hyperbolic as soon as jrj is large enough to ensure that the trace of our appeal to ref.22is the existence of this nondiscrete free τ = a + d + rc is larger than 2 in absolute value. We only need to subgroup, but this existence is much easier to prove directly in the show that a suitable choice of r will ensure the above condition present case of Γ = PSL2(A) than for general nondiscrete 1 on ξ±. Notice that ∞ and g∞ lie in the same component of P \{ξ±}, nonsoluble Γ. because q0 preserves these components and sends ∞ to g∞.In fi conclusion, it suf ces to prove the following two claims: (i)as Remark 11. Here is a direct argumentpffiffiffi avoiding all of the above ref- jrj → ∞,theset{ξ±}convergesto{∞, g∞}; and (ii) changing erences in the examples of A = Z½ 2 or A = Z[1/ℓ], where ℓ is the sign of r (when jrj is large) will change the component of prime. We show directly that the Γ-action on P1 is not amenable. P1 ∞ ∞ ξ \{ , g } in which ± lie (we need it to be the component of gp). We consider Γ as a lattice in L: = PSL2(R) × PSL2(R) in the first The claims can be proved by elementary dynamical considera- case and L: = PSL2(R) × PSL2(Qℓ) in the second case, both times tions; we shall instead verify them explicitly. in such a way that the Γ-action on P1 extends to the L-action x ± 1 The fixed points ξ± are represented by the eigenvectors , factoring through the first factor. If the Γ-action on P were ame- pffiffiffiffiffiffiffiffiffiffiffiffi c where x± = λ± − d − rc and λ ± = ðτ ± τ2 − 4Þ=2 are the eigen- nable, so would be the L-action (by coamenability of the lattice).
Load more

Recommended publications
  • No Tits Alternative for Cellular Automata
    No Tits alternative for cellular automata Ville Salo∗ University of Turku January 30, 2019 Abstract We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alter- native. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group Z) and the one-sided case (spatially acting monoid N, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional sub- shifts of finite type with positive entropy and dense minimal points, auto- morphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field. 1 Introduction In [52] Jacques Tits proved that if F is a field (with no restrictions on charac- teristic), then a finitely-generated subgroup of GL(n, F ) either contains a free group on two generators or contains a solvable subgroup of finite index. We say that a group G satisfies the Tits alternative if whenever H is a finitely-generated subgroup of G, either H is virtually solvable or H contains a free group with two generators. Whether an infinite group satisfies the Tits alternative is one of the natural questions to ask.
    [Show full text]
  • Survey on Geometric Group Theory
    M¨unster J. of Math. 1 (2008), 73–108 M¨unster Journal of Mathematics urn:nbn:de:hbz:6-43529465833 c M¨unster J. of Math. 2008 Survey on geometric group theory Wolfgang L¨uck (Communicated by Linus Kramer) Abstract. This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. Introduction This survey article on geometric group theory is written by a non-expert who likes geometric group theory and uses it in his own research. It is meant as a service for people who want to receive an impression and read an introduction about the topic and possibly will later pass to more elaborate and specialized survey articles or to actual research articles. There will be no proofs. Except for Theorem 7.4 all results have already appeared in the literature. There is to the author’s knowledge no obvious definition what geometric group theory really is. At any rate the basic idea is to pass from a finitely generated group to the geometry underlying its Cayley graph with the word metric. It turns out that only the large scale geometry is really an invariant of the group itself but that this large scale or coarse geometry carries a lot of information. This leads also to a surprising and intriguing variety of new results and structural insights about groups. A possible explanation for this may be that humans have a better intuition when they think in geometric terms.
    [Show full text]
  • Formal Logic Lecture 2
    Faculty of Philosophy Formal Logic Lecture 2 Peter Smith Peter Smith: Formal Logic, Lecture 2 1 Outline Validity again Systematicity and formality Modality and the invalidity principle The counterexample method Peter Smith: Formal Logic, Lecture 2 2 Valid deductions as absolutely watertight inferences I Consider this inference Every good philosopher knows some logic. No post-modernist knows any logic. Jacques is a post-modernist. So: Jacques isn't a good philosopher. Who knows whether the premisses are true or false? But if the premisses are true, the conclusion has to be true too. I The inferential move in this argument from premisses to conclusion is absolutely watertight. There is not even a remote chance, there's no possibility at all, that the premisses are true and the conclusion false. I Such an inference, we said, is deductively valid. I It is conventional to call a (one-step) argument valid if its inferential move is valid. An argument which is valid and has true premisses is called sound. Validity again Logicians' deductions vs Sherlock's deductions I The detective gathers evidence, and `deduces' who did the dastardly deed. I But this (normally) is not a valid deduction in the logician's sense. The detective's account of the murder may fit the facts and strike us as obviously the best explanation. But the best (= most plausible, most likely) explanation doesn't have to be the only one that is logically consistent with the facts. I So the `deduction' from the evidential facts to the detective's proposed explanation isn't absolutely guaranteed to be truth-preserving, i.e.
    [Show full text]
  • Proofs and Mathematical Reasoning
    Proofs and Mathematical Reasoning University of Birmingham Author: Supervisors: Agata Stefanowicz Joe Kyle Michael Grove September 2014 c University of Birmingham 2014 Contents 1 Introduction 6 2 Mathematical language and symbols 6 2.1 Mathematics is a language . .6 2.2 Greek alphabet . .6 2.3 Symbols . .6 2.4 Words in mathematics . .7 3 What is a proof? 9 3.1 Writer versus reader . .9 3.2 Methods of proofs . .9 3.3 Implications and if and only if statements . 10 4 Direct proof 11 4.1 Description of method . 11 4.2 Hard parts? . 11 4.3 Examples . 11 4.4 Fallacious \proofs" . 15 4.5 Counterexamples . 16 5 Proof by cases 17 5.1 Method . 17 5.2 Hard parts? . 17 5.3 Examples of proof by cases . 17 6 Mathematical Induction 19 6.1 Method . 19 6.2 Versions of induction. 19 6.3 Hard parts? . 20 6.4 Examples of mathematical induction . 20 7 Contradiction 26 7.1 Method . 26 7.2 Hard parts? . 26 7.3 Examples of proof by contradiction . 26 8 Contrapositive 29 8.1 Method . 29 8.2 Hard parts? . 29 8.3 Examples . 29 9 Tips 31 9.1 What common mistakes do students make when trying to present the proofs? . 31 9.2 What are the reasons for mistakes? . 32 9.3 Advice to students for writing good proofs . 32 9.4 Friendly reminder . 32 c University of Birmingham 2014 10 Sets 34 10.1 Basics . 34 10.2 Subsets and power sets . 34 10.3 Cardinality and equality .
    [Show full text]
  • Lightning Talk - UNCG Mathematics Summer School 2019
    Lightning Talk - UNCG Mathematics Summer School 2019 Jenny Beck UNC Greensboro June 25, 2019 Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro I First-year PhD student at UNC Greensboro beginning Fall 2019 Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro Jenny Beck - About Me I BS In Mathematics with Pure Mathematics concentration from UNC Greensboro (2014) I MA in Mathematics from UNC Greensboro (2019) I First-year PhD student at UNC Greensboro beginning Fall 2019 Lightning Talk - UNCG Mathematics Summer School 2019 UNC Greensboro Theorem (Tits) A finitely generated linear group is not virtually solvable if and only if it contains a free subgroup. I Historical Context: The Banach-Tarski Paradox and The Von Neumann Conjecture Von Neumann Conjecture (false) A group is non-amenable if and only if it contains a free subgroup I In short, sometimes it pays to be wrong, especially if you’re John Von Neumann.
    [Show full text]
  • Arxiv:1702.07955V1 [Math.GR]
    ABOUT VON NEUMANN’S PROBLEM FOR LOCALLY COMPACT GROUPS FRIEDRICH MARTIN SCHNEIDER Abstract. We note a generalization of Whyte’s geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups. 1. Introduction In his seminal article [18] von Neumann introduced the concept of amenabil- ity for groups in order to explain why the Banach-Tarski paradox occurs only for dimension greater than two. He proved that a group containing an isomorphic copy of the free group F2 on two generators is not amenable. The converse, i.e., the question whether every non-amenable group would have a subgroup being iso- morphic to F2, was first posed in print by Day [5], but became known as the von Neumann problem (or sometimes von Neumann-Day problem). The original ques- tion has been answered in the negative by Ol’ˇsanski˘ı[19]. However, there are very interesting positive solutions to variants of the von Neumann problem in different settings: a geometric solution by Whyte [27], a measure-theoretic solution by Ga- boriau and Lyons [9] and its generalization to locally compact groups by Gheysens and Monod [14], as well as a Baire category solution by Marks and Unger [13]. Whyte’s geometric version reads as follows. Theorem 1.1 (Theorem 6.2 in [27]).
    [Show full text]
  • The Mertens Conjecture
    The Mertens conjecture Herman J.J. te Riele∗ Title of c Higher Education Press This book April 8, 2015 and International Press ALM ??, pp. ?–? Beijing-Boston Abstract The Mertens conjecture states that |M(x)|x−1/2 < 1 for x > 1, where M(x) = P1≤n≤x µ(n) and where µ(n) is the M¨obius function defined by: µ(1) = 1, µ(n) = (−1)k if n is the product of k distinct primes, and µ(n)=0 if n is divisible by a square > 1. The truth of the Mertens conjecture implies the truth of the Riemann hypothesis and the simplicity of the complex zeros of the Riemann zeta function. This paper gives a concise survey of the history and state-of-affairs concerning the Mertens conjecture. Serious doubts concerning this conjecture were raised by Ingham in 1942 [12]. In 1985, the Mertens conjecture was disproved by Odlyzko and Te Riele [23] by making use of the lattice basis reduction algorithm of Lenstra, Lenstra and Lov´asz [19]. The best known results today are that |M(x)|x−1/2 ≥ 1.6383 and there exists an x< exp(1.004 × 1033) for which |M(x)|x−1/2 > 1.0088. 2000 Mathematics Subject Classification: Primary 11-04, 11A15, 11M26, 11Y11, 11Y35 Keywords and Phrases: Mertens conjecture, Riemann hypothesis, zeros of the Riemann zeta function 1 Introduction Let µ(n) be the M¨obius function with values µ(1) = 1, µ(n) = ( 1)k if n is the product of k distinct primes, and µ(n) = 0 if n is divisible by a square− > 1.
    [Show full text]
  • Recursive Formulas Related to the Summation of the Mobius¨ Function
    The Open Mathematics Journal, 2008, 1, 25{34 25 RECURSIVE FORMULAS RELATED TO THE SUMMATION OF THE MOBIUS¨ FUNCTION MANUEL BENITO AND JUAN L. VARONA Abstract. For positive integers n, let µ(n) be the M¨obiusfunction, and M(n) its sum Pn M(n) = k=1 µ(k). We find some identities and recursive formulas for computing M(n); in particular, we present a two-parametric family of recursive formulas. 1. Introduction The well-known M¨obiusfunction µ(n) is defined, for positive integers n, as 8 < 1 if n = 1; µ(n) := (−1)k if n is a product of k different prime numbers, : 0 if there exists a prime p such that p2 divides n (see [1, Chapter 2]). Then, for every real number x ≥ 0, the summation of the M¨obius function is defined by taking bxc X M(x) = M(bxc) := µ(k): k=1 In what follows, and as usually, we refer to M(x) as the Mertens function, although, before F. Mertens (who used it in 1897, see [2]), T. J. Stieltjes already had introduced this function in his attempts to prove the Riemann Hypothesis (see [3, Lettre 79, p. 160{164], dated in 1885). The behaviour of M(x) is rather erratic and difficult of analyze, but it is very important in analytic number theory. In 1912, J. E. Littlewood [4] proved that the Riemann Hypothesis is equivalent to this fact: (1) jM(x)j = O(x1=2+"); when x ! 1; for every " > 0; in relation to this subject, see also [5].
    [Show full text]
  • Variations on a Theme of Borel
    VARIATIONS ON A THEME OF BOREL: AN ESSAY CONCERNING THE ROLE OF THE FUNDAMENTAL GROUP IN RIGIDITY Revision Summer 2019 Shmuel Weinberger University of Chicago Unintentionally Blank 2 Armand Borel William Thurston 3 4 Preface. This essay is a work of historical fiction – the “What if Eleanor Roosevelt could fly?” kind1. The Borel conjecture is a central problem in topology: it asserts the topological rigidity of aspherical manifolds (definitions below!). Borel made his conjecture in a letter to Serre some 65 years ago2, after learning of some work of Mostow on the rigidity of solvmanifolds. We shall re-imagine Borel’s conjecture as being made after Mostow had proved the more famous rigidity theorem that bears his name – the rigidity of hyperbolic manifolds of dimension at least three – as the geometric rigidity of hyperbolic manifolds is stronger than what is true of solvmanifolds, and the geometric picture is clearer. I will consider various related problems in a completely ahistorical order. My motive in all this is to highlight and explain various ideas, especially recurring ideas, that illuminate our (or at least my own) current understanding of this area. Based on the analogy between geometry and topology imagined by Borel, one can make many other conjectures: variations on Borel’s theme. Many, but perhaps not all, of these variants are false and one cannot blame them on Borel. (On several occasions he described feeling lucky that he ducked the bullet and had not conjectured smooth rigidity – a phenomenon indistinguishable to the mathematics of the time from the statement that he did conjecture.) However, even the false variants are false for good reasons and studying these can quite fun (and edifying); all of the problems we consider enrich our understanding of the geometric and analytic properties of manifolds.
    [Show full text]
  • [Math.NT] 6 May 2001 Number Neeti H Imn Yohss Eaddb Aya H M the As Many by Regarded Hypothesis, Riemann the in Interest ABSTRACT
    An Elementary Problem Equivalent to the Riemann Hypothesis Jeffrey C. Lagarias (May 5, 2001 version) n 1 ABSTRACT. The problem is: Let Hn = j be the n-th harmonic number. Show, for each jP=1 n ≥ 1, that d ≤ Hn + exp(Hn) log(Hn), Xd|n with equality only for n = 1. AMS Subject Classification (2000): Primary 11M26, Secondary 11A25 Keywords: Riemann hypothesis, colossally abundant numbers 1. Introduction We consider the following problem. n 1 Problem E. Let Hn = j . Show that, for each n ≥ 1, jP=1 d ≤ Hn + exp(Hn) log(Hn), (1.1) Xd|n with equality only for n = 1. The function σ(n) = d|n d is the sum of divisors function, so for example σ(6) = 12. The number Hn is called the n-thP harmonic number by Knuth, Graham and Patashnik [9, Sect. 6.3], who detail various properties of harmonic numbers. The ‘E’ in Problem E might stand for either ‘easy’ or ‘elementary’. Perhaps ‘H’ for ‘hard’ arXiv:math/0008177v2 [math.NT] 6 May 2001 would be a better letter to use, since our object is to show the following equivalence. Theorem 1.1 Problem E is equivalent to the Riemann hypothesis. The Riemann hypothesis, stated by Riemann [17] in 1859, concerns the complex zeros of the Riemann zeta function. The Riemann zeta function ζ(s) is defined by the Dirichlet series ∞ ζ(s)= n−s, nX=1 which converges for ℜ(s) > 1, and it has an analytic continuation to the complex plane with one singularity, a simple pole with residue 1 at s = 1.
    [Show full text]
  • Arxiv:1812.11743V1 [Math.GR] 31 Dec 2018 Rbe,Wbln Groups
    THE BURNSIDE PROBLEM FOR LOCALLY COMPACT GROUPS THIBAUT DUMONT AND THIBAULT PILLON Abstract. Using topological notions of translation-like actions introduced by Schnei- der, we give a positive answer to a geometric version of Burnside problem for locally compact group. The main theorem states that a locally compact group is non-compact if and only if it admits a translation-like action by the group of integers Z. We then characterize the existence of cocompact translation-like actions of Z or non-abelian free groups on a large class of locally compact groups, improving on Schneider’s results and generalising Seward’s. 1. Introduction Several problems in geometric group theory ask whether properties of a group can be formulated in terms of subgroup containment. The Day-von Neumann problem asks whether a group is non-amenable if and only if it contains a non-abelian free subgroup. The general Burnside problem, which is the simplest form of the problem, asks whether a finitely generated group is infinite if and only if it contains infinite cyclic subgroups. The so-called Gersten conjecture, formulated by Gromov, loosely asks whether any non- hyperbolic group contains a Baumslag-Solitar group BS(m,n). Similarly, a sufficient condition for a group to have exponential growth is to possess a free subsemigroup and it is very natural to wonder if the converse also holds. However, all these questions have been answered by the negative. Ol’shanskii [Oˇs80] proved the existence of non-amenable groups not containing free subgroups. Together with Sapir [OsS02], they extended this result to finitely presented groups and later Monod [Mon13] provided the first explicit examples.
    [Show full text]
  • 9.3.1998 Amenability and Paradoxical
    9.3.1998 AMENABILITY AND PARADOXICAL DECOMPOSITIONS FOR PSEUDOGROUPS AND FOR DISCRETE METRIC SPACES Tullio Ceccherini-Silberstein, Rostislav Grigorchuk and Pierre de la Harpe Abstract. This is an expostion of various aspects of amenability and paradoxical decom- positions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski, according to which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and to defining a Følner condition. Using a Hall-Rado Theorem on matchings in graphs, we show then for pseudogroups that existence of an invariant mean is equivalent to the Følner condition; in the case of the pseudogroup of bounded perturbations of the identity on a discrete metric space, these conditions are moreover equivalent to the negation of the Gromov’s so-called doubling condition, to isoperimetric conditions, to Kesten’s spectral condition for related simple ran- dom walks, and to various other conditions. We define also the minimal Tarski number of paradoxical decompositions associated to a non-amenable group action (an integer ≥ 4), and we indicate numerical estimates (Sections II.4 and IV.2). The final chapter explores for metric spaces the notion of superamenability, due for groups to Rosenblatt. T.C.-S.: Dipartimento di Matematica Pura ed Applicata, Universit`adegli Studi dell’ Aquila, Via Vetoio, I-67100 L’Aquila, Italy E-mail : [email protected] R.G.: Steklov Mathematical Institute, Gubkina Str. 8, Moscow 117 966, Russia. E-mail : [email protected] and [email protected] P.H.: Section de Math´ematiques, C.P.
    [Show full text]