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 $B({\mathcal H}_π)$.
Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amena…
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 $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of finite factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property (T).
We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result proves Kaimanovich's conjecture of the equivalence of amenability and the Liouville property.
We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups, each of which has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with at least one stabilizer of property A. Specializing the…
A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable discrete group is strongly amenable if and only if none of its quotients have the infinite conjugacy class property.
We prove that the action of the automorphism group of a building on its boundary is topologically amenable. The notion of boundary we use was defined in a previous paper \cite{CL}. It follows from this result that such groups have property (A), and thus satisfy the Novikov conjecture. It may also lead to applications in rigidity theory.
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 say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff II_1 factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non implications were obtained in [Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining implications and counterexamples.
Weak amenability of discrete groups was introduced by Haagerup and co-authors in the 1980's. It is an approximation property known to be stable under direct products and free products. In this paper we show that graph products of weakly amenable discrete groups are weakly amenable (with Cowling-Haagerup constant 1). Along the way we construct a wall space associated to the word length structure of a graph product and also give a method for extending completely bounded functions on discrete groups to a completely bounded function on their graph product.
Let $G$ be a subgroup of a discrete (countable) group $Γ$. We introduce a notion of relative inner amenability of $G$ in $Γ$, we prove some equivalent conditions and provide examples as well as counter-examples. We also discuss briefly the corresponding relative property gamma for pairs of finite factors and we deduce from this a characterization of discrete, icc groups which do not have Kazhdan's property $T$.
We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding…
The purpose of this paper is to study the notion of relative extreme amenability for pairs of topological groups. We give a characterization by a fixed point property on universal spaces. In addition we introduce the concepts of an extremely amenable interpolant as well as maximally relatively extremely amenable pairs and give examples. It is shown that relative extreme amenability does not imply the existence of an extremely amenable interpolant. The theory is applied to generalize results of Kechris, Pestov and Todorcevic relating to the application of Fraïssé theory to the theory of Dynamic…
Liouville property of actions of discrete groups can be reformulated in terms of existence co-F$ø$lner sets. Since every action of amenable group is Liouville, the property can be served as an approach for proving non-amenability. The verification of this property is conceptually different than finding a non-amenable action. There are many groups that are defined by strongly transitive actions. In some cases amenability of such groups is an open problem. We define $n$-Liouville property of action to be Liouville property of point-wise action of the group on the sets of cardinality $n$. We refo…
We prove that cocompact (and more generally: undistorted) lattices on $\tilde{A}_2$-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first $\ell^p$-cohomology of an $\tilde{A}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac 4 3,4]$ and the reduced $C^*$-algebra associated to an $\tilde{A}_2$-lattice do not have the operator space approximation property and, consequ…
Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,μ)$ such that $L^\infty(X,μ)$ has an invariant mean and that there is a Borel cocycle $α:G\times X\rightarrow H$ which is proper in a suitable, natural sense. We show that if $H$ has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does $G$. We observe that it is the case for a weak form of measure equivalence for pairs of discrete groups.
We prove that the alternating group of a topologically free action of a countably infinite group $Γ$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $Γ$ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable $Γ$ that there are many such alternating groups which are simple, finitely generated, and C$^*$-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance o…
This paper is devoted to the study of universality for a particular continuous action naturally attached to certain pairs of closed subgroups of $S_{\infty}$. It shows that three new concepts, respectively called relative extreme amenability, relative Ramsey property for embeddings, and relative Ramsey property for structures, are relevant in order to understand this property correctly. It also allows to provide a partial answer to a question posed by Kechris, Pestov and Todorcevic.
A finite von Neumann algebra $\mathcal{M}$ with a faithful normal trace $% τ$ has Haagerup's approximation property (relative to a von Neumann subalgebra $\mathcal{N}$) if there exists a net $(ϕ_α)_{α\in Λ}$ of normal completely positive ($\mathcal{N}$-bimodular) maps from $\mathcal{M}$ to $\mathcal{M}$ that satisfy the subtracial condition $% τ\circ ϕ_α\leq τ$, the extension operators $% T_{ϕ_α}$ are bounded compact operators (in $<\mathcal{M%},e_{\mathcal{N}}>$), and pointwise approximate the identity in the trace-norm, i.e., $\lim_α||ϕ_α(x)-x||_{2}=0$ for all $% x\in \mathcal{M}$. We prove …
In this paper we prove that the general version, F(N) of the Thompson group is inner amenable. As a consequence we generalize a result of P.Jolissaint. To do so, we prove first that F(N) together with a normal subgroup are i.c.c (infinite conjugacy classes) groups. Then, we investigate the relative McDuff property out of which we extract property $Γ$ for the group von Neumann algebras involved. By a result of E.G.Effros, F(N) follows inner amenable.
In this paper we prove that the Thompson groups $T$ and $V$ are not inner amenable. In particular, their group von Neumann algebras do not have property $Γ$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is simple, then the Thompson group $F$ is non-amenable. Furthermore, we give a few new equivalent characterizations of amenability of $F$.
Recently Akhmedov introduced a property of finitely generated groups called the Traveller Salesman Property (TSP). It was shown by Thurston that TSP implies non-amenability. Akhmedov conjectured that R. Thompson's group $F$ has TSP. The aim of this article is to disprove this conjecture.
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…
We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem. We study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X. Since relative amenability is closed under Chabauty limits, it follows that any Chabauty l…
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…
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…
A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for transformation groupoids relative to actions of exact discrete groups. We believe that for general étale groupoids one should have the same equivalence of the two properties under some mild exactness assumption. In this paper we try to support this statement.
The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author and the Haagerup property introduced by Connes and Choda. In this paper it is shown that a connected simple Lie group G has the weak Haagerup property if and only if the real rank of G is zero or one. Hence for connected simple Lie groups the weak Haagerup property coincides with weak amenability. Moreover, it turns out that for connected simple Lie groups the…
We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin, our result recovers that mapping class groups and Out(F_n) are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann.
The original definition of amenability given by von Neumann in the highly non-constructive terms of means was later recast by Day using approximately invariant probability measures. Moreover, as it was conjectured by Furstenberg and proved by Kaimanovich-Vershik and Rosenblatt, the amenability of a locally compact group is actually equivalent to the existence of a single probability measure on the group with the property that the sequence of its convolution powers is asymptotically invariant. In the present article we extend this characterization of amenability to measured groupoids. It implie…
The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $σ$-finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.
We introduce two natural notions of cogrowth for finitely generated semigroups --- one local and one global --- and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when d…
We observe that a Polish group $G$ is amenable if and only if every continuous action of $G$ on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as amenability at infinity of discrete countable groups. As corollary, the latter property can also be tested by actions on the Hilbert cube. These results generalize a criterion due to Giordano and de la Harpe.
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 $X$.
Let $Γ$ be a countable group acting on a countable set $X$ by permutations. We give a necessary and sufficient condition for the action to have a quasi-invariant mean with a given cocycle. This can be viewed as a combinatorial analogue of the condition for the existence of a quasi-invariant measure in the Borel case given by Miller. Then we show a geometric condition that guarantees that the corresponding action on the Stone-Čech compactification is Zimmer amenable. The geometric condition (weighted hyperfiniteness) resembles Property A. We do not know the exact relation between the two notion…
A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability.
We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer …
The Haagerup property is not invariant under quasi-isometry
Rockestate SRL；University of Illinois at Chicago（芝加哥伊利诺伊大学）；Haute école d’Ingénierie et de Gestion du Canton de Vaud（沃州高等工程与管理学院）；Institut de Mathématiques, UniMail（洛桑大学数学研究所）
We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.
In this thesis, we study the existence of universal objets of two differents types in the theory of topological groups and theirs actions on compacts spaces. In the first part, we contribute to the problem of existence of test spaces for amenability. We observe that a Polish group is amenable if and only if its every continuous action on the Hilbert cube admits an invariant probability measure. This generalizes a result of Bogatyi and Fedorchuk. We also show that actions on the Cantor space can be used to detect amenability and extreme amenability of Polish non-archimedean groups as well as am…
We prove an extension property for $M_d$-multipliers from a subgroup to the ambient group, showing that $M_{d+1}(G)$ is strictly contained in $M_d(G)$ whenever $G$ contains a free subgroup. Another consequence of this result is the stability of the $M_d$-approximation property under group extensions. We also show that Baumslag-Solitar groups are $M_d$-weakly amenable with $\boldsymbolΛ(\operatorname{BS}(m,n),d)=1$ for all $d\geq 2$. Finally, we show that, for simple Lie groups with finite centre, $M_d$-weak amenability is equivalent to weak amenability, and we provide some estimates on the con…
We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk's group, Basilica group, the full topological group of Cantor minimal system, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, groups that naturally appear in holomorph…
This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity" from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a si…
In his work on the Novikov conjecture, Yu introduced Property $A$ as a readily verified criterion implying coarse embeddability. Studied subsequently as a property in its own right, Property $A$ for a discrete group is known to be equivalent to exactness of the reduced group $C^*$-algebra and to the amenability of the action of the group on its Stone-Cech compactification. In this paper we study exactness for groups acting on a finite dimensional $\CAT(0)$ cube complex. We apply our methods to show that Artin groups of type FC are exact. While many discrete groups are known to be exact the que…
We show that amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking second nilpotent product of groups. We also define the restricted second nilpotent wreath product of groups, this is a semi-direct product akin to the restricted wreath product but constructed from the second nilpotent product. We then show that if two discrete groups have the Haagerup property, the restricted second nilpotent wreath product of them also has the Haagerup property. We finally show that if a discrete group is abelian, then the restricted second nilpotent wreath pr…
Let $\cF$ be a family of finite loops closed under subloops and factor loops. Then every loop in $\cF$ has the strong Lagrange property if and only if every simple loop in $\cF$ has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops.
