Questions tagged [amenability]

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12 votes
1 answer
373 views

Does every topological group embed as a closed subgroup in an amenable group?

It is a standard result that closed subgroups of locally compact amenable groups are themselves amenable, so for example $F_2$, the free group on two generators, cannot be embedded as a closed ...
1 vote
0 answers
145 views

Does every amenable group $G$ admit a two-sided Folner sequence?

By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence. Context: I just came up with this question and surprisingly I haven'...
0 votes
0 answers
73 views

Sequential approximate diagonal

Let $A$ be a unital, amenable Banach algebra. What is the significance of $A$ to have a weakly Cauchy sequential approximate diagonal? A preliminary observation: Let $\displaystyle A\hat{\otimes}_{\...
0 votes
0 answers
90 views

Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$

Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
5 votes
3 answers
395 views

Examples of amenable Banach algebras which have non-amenable subalgebra

I am looking for examples of amenable Banach algebras which have non-amenable subalgebra I know 1: Each amenable Banach algebra has a bounded approximate identity 2: If $I$ be a closed ideal in an ...
1 vote
0 answers
99 views

Amenability of $\textrm{w}_0(L^1(G))$

Let $G$ be an infinite compact group and $A=L^1(G)$. It is known that $c_0(A)$ is amenable [Runde2020, p.80] while $\ell^{\infty}(A)$ is not [Daws2009] . Let $\textrm{w}_0(A)$ denote the subspace of $\...
3 votes
1 answer
134 views

Topological amenability of actions - forgetting topology

Let $G$ be a (countable) discrete group and let $X$ be a locally compact Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Recall that the action is (topologically) amenable if there ...
14 votes
2 answers
504 views

Subgroups of amenable periodic groups

Does every countable, infinite, amenable, periodic group $G$ contain an infinite locally finite subgroup? Remarks: I would be happy with an infinitely generated counterexample as long as it is ...
15 votes
1 answer
898 views

Is the infinite product of solvable groups amenable?

I am interested in the amenability properties of infinite products of solvable groups. The following facts are well-known: Any solvable group is amenable. The class of solvable groups is closed under ...
1 vote
1 answer
99 views

Nonamenable p.m.p. action on a standard probability space

Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations. Is the action of $G$ always amenable? (Amenable action, ...
5 votes
1 answer
227 views

Extreme amenability of topological groups and invariant means

Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ...
2 votes
0 answers
80 views

Property A, Higson-Roe condition and its applications

Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition: Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...
3 votes
1 answer
135 views

Kahler groups with no non-abelian free groups?

There are well-known results about nilpotent and solvable (=virtually nilpotent) Kähler groups coming from the work of (to name a few) Campana, Carlson-Toledo, Arapura-Nori, Delzant... Are there any ...
10 votes
1 answer
239 views

Faithful extreme traces on group C*-algebras

Let $G$ be a discrete amenable, residually finite, ICC(i.e. each non-trivial conjugacy class is infinite) group. Let $C^*_r(G)$ be the reduced group $C^*$-algebra of $G$. Since $G$ is ICC the (...
9 votes
3 answers
499 views

Spectral radius of a finitely generated group

Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
9 votes
1 answer
713 views

Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...
7 votes
1 answer
153 views

Density of “diagonal sets” in amenable groups

Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that $$ \lim_{n \to \infty} \...
4 votes
1 answer
500 views

Amenable subsets of groups

Let $X$ be a subset of a group $G$. We say that $X$ is left amenable with respect to $G$ if there is a function $\mu:\mathcal P(G)\to [0,\infty]$ with the following three properties. $\mu(A\cup B)=\...
-1 votes
1 answer
150 views

Følner sequences of the integers

Definition: Let $G$ be a group. For $g\in G$ and a subset $F\subseteq G$ fix the notation $gF:=\{gf\mid f\in G\}$. A sequence $(F_{i})_{i\in\mathbb{N}}\subseteq G$ is called a Følner sequence if \...
2 votes
1 answer
226 views

Paradoxical decomposition modulo finite sets

Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there ...
11 votes
1 answer
552 views

If $(F_n)_n$ is a Følner sequence satisfying Tempelman's condition, is $(F_n^{-1}F_n)_n$ also Følner?

Let $G$ be a countable group. A Følner sequence is a sequence of finite subsets $(F_n)_n$ such that $$\lim_{n\to\infty} \frac{|KF_n \mathbin\triangle F_n|}{|F_n|} = 0$$ for each fixed finite subset $K ...
3 votes
2 answers
254 views

Upper density of subsets of an amenable group

Let $G$ be an amenable group (so locally compact Hausdorff) and also assume it is second countable if needed. My question is that what are the standard ways (across literature) of defining the upper ...
9 votes
2 answers
3k views

Characterization of amenable actions

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
7 votes
2 answers
514 views

The kernel of all invariant means

Let $G$ be a discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g \in ...
11 votes
0 answers
244 views

Minimal actions commuting with amenable actions of $\mathbb{F}_2$

For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
1 vote
0 answers
105 views

Closed subgroups of totally disconnected Polish amenable groups

Let $G$ be a totally disconnected Polish topological group (e.g., a closed subgroup of the homeomorphism group of the Cantor set). If $G$ is amenable, is every closed subgroup of $G$ also amenable? ...
1 vote
0 answers
165 views

Closure of an amenable subgroup

Let $G$ be a topological group, and let $H < G$ be a countable subgroup that is amenable as a discrete group. Is the closure of $H$ an amenable topological group?
3 votes
0 answers
244 views

Amenability, growth and asymptotic dimension

I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...
9 votes
1 answer
298 views

Do extensions of pure states separate points?

Let $B$ be a unital C*-algebra and let $A⊆B$ be a closed *-subalgebra containing the unit of $B$. I am mostly interested in the case that $A$ is abelian but, for the strict purpose of stating my ...
6 votes
1 answer
293 views

Trans-amenability of group actions

This problem is derived from this post. Let $G$ be a countable discrete group and $H\le G$ be a subgroup. Consider the $G$-action on $X=G/H$. Then the following amenability-like conditions are ...
8 votes
0 answers
163 views

Uniform amenability at infinity

Let's recall that a group $G$ is amenable if for any finite subset $E\subset G$ and any $\epsilon>0$ there is a finite subset $F\subset G$ such that $$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
2 votes
2 answers
517 views

Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution

Let $G$ be a discrete group. Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution? I only know Johnson's theorem: A group is amenable if and only if the ...
7 votes
2 answers
759 views

Amenable action intuition

Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
38 votes
1 answer
3k views

Is there any version of the Banach-Tarski paradox in ZF?

The Banach-Tarski paradox states that for a solid ball in 3‑dimensional space, there exists a decomposition into a finite number of disjoint subsets, which can then be put back together in a different ...
16 votes
3 answers
1k views

What are the main open problems in the theory of amenability of groups?

I have read the Paterson and Runde books about amenability of groups, but I do not know what are the most intriguing questions in this area today. A survey or a list of questions would be welcome.
87 votes
1 answer
9k views

Non-amenable groups with arbitrarily large Tarski number?

Just out of curiosity, I wonder whether there are non-amenable groups with arbitrarily large Tarski numbers. The Tarski number $\tau(G)$ of a discrete group $G$ is the smallest $n$ such that $G$ ...
8 votes
0 answers
148 views

Amenable automatic groups

Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if ...
6 votes
1 answer
220 views

Introductory text on amenability

I am looking for a book that covers amenability rigorously. Preferably a book aimed at beginners.
7 votes
0 answers
127 views

Can a lacunary hyperbolic group be Liouville?

While discussing with a colleague of a possible use of lacunary hyperbolic elementarily amenable groups introduced by Olshanskii, Osin & Sapir, it occurred to me that I was not aware whether ...
2 votes
1 answer
122 views

Is it possible to characterize the elements of the C$^*$-algebra of an open subgroupoid?

$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$Let $\G$ be an etale groupoid, let $U$ be an open subset of $\G^{(0)}$, and let $$ \H = \{\gamma \in \G:...
4 votes
0 answers
155 views

Is the group ring of an amenable group, viewed as multiplicative monoid, amenable?

Motivated by this question, it seems natural to ask the following: Question 1: Is there a [finitely generated discrete] torsion-free virtually Abelian (but not Abelian) group $G$ so that the ...
8 votes
1 answer
321 views

Amenable groupoid C*-algebras satisfy the UCT in English?

As is by now well known, Tu proved in 1998 that the C*-algebras coming from amenable groupoids satisfy the so-called UCT (universal coefficient theorem). Unfortunately, I don't speak french and I've ...
3 votes
3 answers
477 views

Følner sequences with weird shapes

Let $G$ be a discrete and finitely generated group. Recall that $\{F_n\}_{n \in \mathbb{N}}$ is a Følner sequence if $|g F_n \cup F_n|/|F_n| \rightarrow 1$ for every $g \in G$. As is well known, ...
10 votes
3 answers
844 views

Explicit free subgroup in Thompson's group $V$

R. Thompson introduced three groups $F\subset T\subset V$. The question concerning amenability of $F$ is still unanswered and has attracted much attention. I have read that Thompson group $V$ contains ...
0 votes
1 answer
99 views

Is there a $G$-paradoxical $G$-invariant subset of the plane for $G$ a group of rigid motions?

The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $...
0 votes
2 answers
209 views

Induced probability measure on a finite orbit under a group action

Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$ via measure-preserving homeomorphisms, and suppose we have a point $x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
4 votes
3 answers
506 views

Amenable Thompson-like groups

Question: Do there exist amenable Thompson-like groups? I realise that my question is vague, but defining and studying groups which look like Thompson's groups $F$, $T$ and $V$ seems to be an ...
6 votes
2 answers
777 views

Is the set of all ICC amenable groups countable?

Is the set of all ICC amenable groups countable? If "yes", then in general, the classes of all countable ICC groups that give rise to the same von Neumann algebra (factor) -- are these ...
1 vote
1 answer
143 views

What is the relation between the Tarski number and growth of a group?

When I was studying the structure of the Grigorchuk group, a question came to my mind and I just had the following information: We know that every finitely generated group of subexponential growth is ...
1 vote
1 answer
68 views

Invariant strictly positive hyperreal probability measures on groups

Under what conditions is there a strictly positive hyperreal probability measure on a group $G$? This would be a finitely-additive non-negative function $\mu$ from the powerset of $G$ to a hyperreal ...