Related papers: Non-inner amenability of the Higman-Thompson group…

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 $\Gamma$. Moreover, we prove that if the reduced group $C^\ast$-algebra of $T$ is…

We prove that, if a discrete group $G$ is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of $G$ is non-amenable with respect to the topology generated by its rank metric. This…

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…

The purpose of this article is prove that Thompson's group F is amenable. The methods developed will then be used to prove a generalization of Hindman's theorem for the free nonassociative binary system on one generator.

Richard Thompson's group F is the group of piecewise linear homeomorphisms of the unit interval with a finite number of break points, all at dyadic rational numbers (their denominators are powers of 2) and with slopes which are powers of 2.…

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in…

A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$…

We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite…

We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group $G$ are similar to unitary ones, is the group amenable? In \S 5, we give a new proof of…

We prove that weak amenability of a locally compact group imposes a strong condition on its amenable closed normal subgroups. This extends non weak amenability results of Haagerup (1988) and Ozawa--Popa (2010). A von Neumann algebra…

In this paper we generalize techniques of Belk-Matucci to solve the conjugacy problem for every Thompson-like group $V_n(H)$, where $n \geq 2$ and $H$ is a subgroup of the symmetric group on $n$ elements. We use this to prove that, if $n…

We provide an example of two elementarily equivalent countable ICC groups $G$ and $H$ such that $G$ is amenable and $H$ is not inner amenable. As a result, we provide the first example of elementarily equivalent groups whose group von…

We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that 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…

We consider the universal Grigorchuk 2-group, i.e., the group such that every Grigorchuk 2-group is a quotient. We show that this group has a nice universal representation in the group of all functions f:{0,1,2}^N --> Aut(T_2), where T_2 is…

Let $F$ denote the Thompson group with standard generators $A=x_0$, $B=x_1$. It is a long standing open problem whether $F$ is an amenable group. By a result of Kesten from 1959, amenability of $F$ is equivalent to $$(i)\qquad ||I+A+B||=3$$…

We prove non-amenability of the product replacement graphs \Gamma_n(G) for uniformly non-amenable groups. We also prove it for Z-large groups, when n is sufficiently large. It follows that \Gamma_n(G) is non-amenable when n is sufficiently…

The purpose of this article is to formulate conjectural generalizations of Hindman's Theorem and Ellis's Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson's group $F$. Partial results are…

It is a long standing open problem whether the Thompson group $F$ is an amenable group. In this paper we show that if $A$, $B$, $C$ denote the standard generators of Thompson group $T$ and $D:=C B A^{-1}$ then…

Let $M$ be a finite von Neumann algebra (resp. a type II$_{1}$ factor) and let $N\subset M$ be a II$_{1}$ factor (resp. $N\subset M$ have an atomic part). We prove that the inclusion $N\subset M$ is amenable implies the identity map on $M$…