We show that free Burnside groups of sufficiently large odd exponent are non--amenable in a certain strong sense, more precisely, their left regular representations are isolated from the trivial representation uniformly on finite generating sets. This result is applied to the solution of a strong version of the von Neumann -- Day problem concerning amenability of groups without non--abelian free subgroups. As another consequence, we obtain that the above--mentioned groups are of uniform exponential growth.
By studying the so-called traveling salesman groups, we obtain a new metric criterion for non-amenability. As an application, we give a new and very short proof of non-amenability of free Burnside groups with sufficiently big odd exponent.
J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.
For a finitely generated group $G$ the Nielsen graph $N_n(G)$, $n\geq \operatorname{rank}(G)$, describes the action of the group $\operatorname{Aut}F_n$ of automorphisms of the free group $F_n$ on generating $n$-tuples of G by elementary Nielsen moves. The question of (non)amenability of Nielsen graphs is of particular interest in relation with the open question about Property $(T)$ for $\operatorname{Aut}F_n$, $n\geq 4$. We prove nonamenability of Nielsen graphs $N_n(G)$ for all $n\ge \max\{2,\operatorname{rank}(G)\}$ when $G$ is indicable, and for $n$ big enough when $G$ is elementary amenab…
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
In this paper, we provide several characterisations for uniform amenability concerning a family of finitely generated groups. More precisely, we show that the Hulanicki-Reiter condition for uniform amenability can be weakened in several directions, including cardinalities of supports and certain operator norms.
We introduce a model for random groups in varieties of $n$-periodic groups as $n$-periodic quotients of triangular random groups. We show that for an explicit $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities $d\in(1/3,d_{\mathrm{crit}})$ and for $n$ large enough, the model produces \emph{infinite} $n$-periodic groups. As an application, we obtain, for every fixed large enough $n$, for every $p\in (1,\infty)$ an infinite $n$-periodic group with fixed points for all isometric actions on $L^p$-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindricall…
The main results of this paper show that various coarse (`large scale') geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated. The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some `bad' spaces. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.
We prove that a uniformly coarsely proper hyperbolic cone over a bounded metric space consisting of a finite union of uniformly coarsely connected components each containing at least two points is non-amenable and apply this to visual Gromov hyperbolic spaces.
The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, there are some classes of automata that do not contain such examples. For instance, all infinite Burnside automaton groups in the literature are generated by non reversible Mealy automata and it was recently shown that 2-state invertible-reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state invertible-reversible Mealy automata, using new original techniques. The results provide the first uniform method to construct elements …
We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number. Furthermore, we show that the established inequality yields an exact equation for perfect Hausdorff dynamical systems. Utilizing these results, we conclude that a dynamical system is amenable if its asymptotic uniform complexity equals one, and that the converse is true for perfect Hausdorff dynamical systems, such as non-discrete Hausdorff topological groups…
We show that the groups of finite energy loops and paths (that is, those of Sobolev class $H^1$) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation $π$ of such a group (which we call skew-amenable) has a conjugation-invariant state on $B({\mathcal H}_π)$.
We prove non-amenability of the product replacement graphs Γ_n(G) for uniformly non-amenable groups. We also prove it for Z-large groups, when n is sufficiently large. It follows that Γ_n(G) is non-amenable when n is sufficiently large for hyperbolic groups, linear groups, and elementary amenable groups.
We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition -- the latter even with regard to arbitrary uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the considered metric Fraïssé class has a ce…
For locally compact groups amenability and Kazhdan's property (T) are mutually exclusive in the sense that a group having both properties is compact. This is no longer true for more general Polish groups. However, a weaker result still holds for SIN groups (topological groups admitting a basis of conjugation-invariant neighbourhoods of identity): if such a group admits sufficiently many unitary representations, then it is precompact as soon as it is amenable and has the strong property (T) (i.e. admits a finite Kazhdan set). If an amenable topological group with property (T) admits a faithful …
Let (G,S) be a Coxeter group. We construct a continuation, to the open unit disc, of the unitary representations associated to the positive definite functions g -> r^l(g). (Here 0<r<1, and l denotes the length function with respect to the generating set S.) The constructed representations are uniformly bounded and we prove that this implies the weak amenability of the group G.
Generalizing Block and Weinberger's characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By considering the case of the natural action of $G$ on its Stone-\vCech compactification we obtain a homological characterization of exactness of the group, answering a question of Nigel Higson.
In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph…
We prove that if G is a discrete group that admits a metrically proper action on a finite-dimensional CAT(0) cube complex X, then G is weakly amenable. We do this by constructing uniformly bounded Hilbert space representations for which the quantities z^{l(g)} are matrix coefficients. Here l is a length function on G obtained from the combinatorial distance function on the complex X.
We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r^d have uniformly bounded norms for 0<r<1. We then combine this with a Bozejko-Picardello type inequality to obtain weak amenability.
Motivated by the Benjamini-Schramm non-unicity of percolation conjecture we study the following question. For a given finitely generated non-amenable group $Γ$, does there exist a generating set $S$ such that the Cayley graph $(Γ,S)$, without loops and multiple edges, has non-unique percolation, i.e., $p_c(Γ,S)<p_u(Γ,S)$? We show that this is true if $Γ$ contains an infinite normal subgroup $N$ such that $Γ/ N$ is non-amenable. Moreover for any finitely generated group $G$ containing $Γ$ there exists a generating set $S'$ of $G$ such that $p_c(G,S')<p_u(G,S')$. In particular this applies to fr…
It is shown that $2β_1(\G)\leq h(\G)$ for any countable group $\G$, where $β_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\ell^2$-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type $\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2β_1(R)\leq 2C(R)-2\leq h(R), $$ where $β_1(\R)$ is the first $\ell…
Given a closed Riemannian manifold M and a (virtual) epimorphism from the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M_n}_{n=0}^{\infty} of M such that the first non-zero eigenvalues λ_1(M_n) of the Laplacian converge to zero as n tends to infinity. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.
We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of thi…
A group $Γ$ is said to be uniformly HS stable if any map $φ: Γ\to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability.
We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.
In this article we introduce and study uniform and non-uniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasi-crystals (a.k.a. Meyer sets) in lcsc abelian groups. We show that envelopes of strong approximate lattices are unimodular, and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor-Sc…
We propose a new look at the Julg-Valette theorem on K-theoretic amenability for groups operating on trees. The main tool is a generalization of a construction of uniformly bounded representations of the free groups due to Pytlik and Szwarc.
We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain "weak Bohr compactification" introduced in [24]. Formally, $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjec…
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{ω_1 ω}$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property {\bf (T)} can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.
We extend Følner's amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte's geometric solution to the von Neumann problem and provide an affirmative answer to a question posed by Rosendal.
