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12 votes
2 answers
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Group with a translation invariant ultrafilter

Let $G$ be an infinite, discrete, countable group. Can $G$ have a translation-invariant ultrafilter? An ultrafilter $\mathcal{F} \subset 2^G$ is translation-invariant if $A \in \mathcal{F}$ implies $g ...
Vladimir's user avatar
  • 1,322
2 votes
0 answers
79 views

Amenability and the unitary group of an operator algebra

Let $M$ be a von Neumann algebra and $U(M)=\{x\in M: x^*=x^{-1}\}$ be its unitary group. In this post, we equip $U(M)$ only with the relative weak$^*$ topology $\sigma(M,M_*)$. Then, $U(M)$ is a ...
Onur Oktay's user avatar
  • 2,283
2 votes
0 answers
93 views

Can we find background noise for every Følner sequence in a countable amenable group?

Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$. I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner ...
Saúl RM's user avatar
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4 votes
0 answers
92 views

Non-monotileable amenable groups

This is crossposted from MSE. We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss ...
Saúl RM's user avatar
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1 vote
0 answers
153 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'...
Saúl RM's user avatar
  • 10.3k
0 votes
0 answers
74 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}_{\...
Onur Oktay's user avatar
  • 2,283
0 votes
0 answers
92 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 ...
Onur Oktay's user avatar
  • 2,283
1 vote
0 answers
100 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 $\...
Onur Oktay's user avatar
  • 2,283
3 votes
1 answer
137 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 ...
Alcides Buss's user avatar
15 votes
1 answer
927 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 ...
Asgar's user avatar
  • 153
1 vote
1 answer
114 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, ...
Ujan Chakraborty's user avatar
5 votes
1 answer
242 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 ...
Muduri's user avatar
  • 225
2 votes
0 answers
82 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 $A$ ...
Ken's user avatar
  • 21
3 votes
1 answer
138 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 ...
Nati's user avatar
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10 votes
1 answer
254 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 (...
Caleb Eckhardt's user avatar
-1 votes
1 answer
157 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 \...
worldreporter's user avatar
4 votes
1 answer
501 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)=\...
Karl Lorensen's user avatar
11 votes
1 answer
569 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 ...
Rob's user avatar
  • 213
2 votes
1 answer
229 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 ...
Alexander Pruss's user avatar
7 votes
1 answer
154 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} \...
Adam's user avatar
  • 323
9 votes
3 answers
520 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 ...
SMS's user avatar
  • 1,335
11 votes
0 answers
247 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 ...
Shirly Geffen's user avatar
1 vote
0 answers
112 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? ...
Vladimir's user avatar
  • 1,322
1 vote
0 answers
170 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?
Vladimir's user avatar
  • 1,322
3 votes
0 answers
271 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 ...
Just dropped in's user avatar
9 votes
1 answer
319 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 ...
Ruy's user avatar
  • 2,233
6 votes
1 answer
298 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 ...
Narutaka OZAWA's user avatar
8 votes
0 answers
170 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 \...
Narutaka OZAWA's user avatar
7 votes
2 answers
790 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 ...
Andromeda's user avatar
  • 169
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 ...
mahdi meisami's user avatar
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.
8 votes
0 answers
155 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 ...
Matt Zaremsky's user avatar
7 votes
0 answers
133 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 ...
ARG's user avatar
  • 4,352
2 votes
1 answer
127 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:...
Ruy's user avatar
  • 2,233
4 votes
0 answers
156 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 ...
ARG's user avatar
  • 4,352
8 votes
1 answer
330 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 ...
Diego Martinez's user avatar
3 votes
3 answers
512 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, ...
Diego Martinez's user avatar
0 votes
1 answer
101 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 $...
Alexander Pruss's user avatar
12 votes
1 answer
380 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 ...
Alessandro Codenotti's user avatar
0 votes
2 answers
213 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 $|Gx| = n$...
James Propp's user avatar
  • 19.5k
6 votes
2 answers
799 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 ...
Chilperic's user avatar
  • 111
1 vote
1 answer
71 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 ...
Alexander Pruss's user avatar
1 vote
1 answer
144 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 ...
mahdi meisami's user avatar
2 votes
0 answers
132 views

About Tarski number 7

Recall that a group $G$ admits a $(m+n)$-paradoxical decomposition if there exist positive integers $m$ and $n$, a partition $\{P_1,\dotsc,P_m,Q_1,\dotsc,Q_n\}$ of $G$ and elements $x_i, y_j$ of $G$ ...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
239 views

Latest progress on Tarski numbers

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...
Meisam Soleimani Malekan's user avatar
5 votes
3 answers
400 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 ...
user62498's user avatar
  • 813
3 votes
2 answers
266 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 ...
Otto's user avatar
  • 1,006
6 votes
1 answer
225 views

Introductory text on amenability

I am looking for a book that covers amenability rigorously. Preferably a book aimed at beginners.
Yiftach Barnea's user avatar
2 votes
0 answers
88 views

Measure invariant under circle maps

Consider continuous bijections (may even assume these are homeomorphisms or diffeomorphisms if it helps) from the circle onto itself given by $x \mapsto x + s_i(x)$ where $i = 1,2$ or $3$. (I'm ...
ARG's user avatar
  • 4,352
8 votes
2 answers
432 views

Constant Martin kernel and amenability

Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
Klaus Thomsen's user avatar