Questions tagged [amenability]

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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 $G$ ...
Narutaka OZAWA's user avatar
64 votes
7 answers
14k views

Is Thompson's Group F amenable?

Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ...
ADL's user avatar
  • 2,752
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
21 votes
1 answer
2k views

Amenability of groups

Let $G$ be non-amenable finitely generated group. 1) Is it true that there exists a sequence $S(n)$ of sets which generate $G$ and such that $\frac{1}{|S(n)|}||\sum_{g\in S(n)} \lambda(g)||\...
Kate Juschenko's user avatar
21 votes
0 answers
574 views

The multiplication game on the free group

Fix $W\subseteq\mathbb F_2$ and consider the following two-person game: Player 1 and Player 2 simultaneously choose $x$ and $y$ in $\mathbb F_2$. The first player wins, say one dollar, iff $xy\in W$. ...
Valerio Capraro's user avatar
20 votes
2 answers
1k views

Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice

Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals. More formally does ...
Josh F's user avatar
  • 545
20 votes
1 answer
1k views

What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$. The problem seems to generate both proofs and disproofs at a fairly high rate, ...
Tobias Kildetoft's user avatar
18 votes
0 answers
544 views

Do quotients of amenable groups C*-algebras satisfy the UCT?

Let G be a discrete amenable group. General Question: Let $J$ be an ideal of $C^*(G)$, the group C*-algebra of $G.$ Does $C^*(G)/J$ satisfy the universal coefficient theorem (UCT)? I am mainly ...
Caleb Eckhardt's user avatar
16 votes
3 answers
640 views

Is the isomorphism problem for amenable groups decidable?

Is it algorithmically decidable if two finitely presented amenable groups are isomorphic? Or slightly different: Does there exist a family of amenable groups (indexed by natural numbers) for which ...
Hans's user avatar
  • 261
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.
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 ...
Asgar's user avatar
  • 153
15 votes
3 answers
2k views

Alternative proofs of the Krylov-Bogolioubov theorem

The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...
Ian Morris's user avatar
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15 votes
1 answer
978 views

Amenable groups with finite classifying space

A group $G$ is said to be elementary amenable if it can be obtained from finite and abelian groups by subgroups, quotients, extensions and increasing unions. It is well-known that all such groups are ...
Andreas Thom's user avatar
  • 25.3k
15 votes
1 answer
683 views

Amenability of groups in terms of a perturbation condition

Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$ $$\inf \lbrace\...
Andreas Thom's user avatar
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14 votes
1 answer
1k views

Amenability of groups II

Are there any non-amenable group $G$ with the property: There exists $C<1$ such that for every finite set $S\subset G$ there exists a set $F\subseteq S$ such that $|F|\geq C |S|$ and $F$ generates ...
Kate Juschenko's user avatar
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 ...
Igor Belegradek's user avatar
14 votes
0 answers
618 views

Some questions on unitarisability of discrete groups

In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible. A discrete group $G$ is unitarisable if for every Hilbert ...
Kate Juschenko's user avatar
12 votes
1 answer
634 views

Is there a notion of 'amenable ring'

Amenable groups are everywhere these days, as examples of all kinds of lovely phenomena. And there are various ways of defining notions of 'amenable monoid' or possibly 'amenable semigroup'. But for ...
Peter Kropholler's user avatar
12 votes
2 answers
357 views

Are finitely generated amenable groups positively finitely generated?

Let $G$ be a finitely generated amenable group. Is there a positive integer $n$ such that $n$ random elements of $G$ generate it with positive probability? Being more formal, note that $G^n$ is ...
Pablo's user avatar
  • 11.2k
12 votes
1 answer
470 views

Topological amenability vs amenability of an action

Let $G$ be a discrete group and let $X$ be a compact, Hausdorff space. Assume that $G$ acts on $X$ by homeomorphisms. Consider the following two definitions: [$C^*$-algebras and finite dimensional ...
13829's user avatar
  • 121
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 ...
Alessandro Codenotti's user avatar
12 votes
0 answers
269 views

Star-shaped Folner sequence

Fix a (finite) generating set $S$ for $\Gamma$ (discrete) amenable. Given a Følner sequence (i.e. a sequence of finite sets $F_n$ whose boundary $\partial F_n$ in the Cayley graph of $S$ is such that $...
ARG's user avatar
  • 4,342
11 votes
1 answer
553 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
11 votes
1 answer
295 views

Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF (...
user56097's user avatar
  • 402
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 ...
Shirly Geffen's user avatar
11 votes
0 answers
216 views

Proving amenability of an extension by using paradoxical decompositions

It is well known that an extension of an amenable group by an amenable group is amenable. Is it possible to prove that by using only paradoxical decompositions: if $G$ has a paradoxical decomposition ...
user avatar
10 votes
4 answers
2k views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
Kamran Reihani's user avatar
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 ...
John N.'s user avatar
  • 723
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 (...
Caleb Eckhardt's user avatar
10 votes
1 answer
229 views

Naturally occurring, non-amenable Zappa-Szep products of discrete amenable groups?

We say $G$ is the Zappa-Szep product of two subgroups $K$ and $P$ if $K\cap P = \{e\}$ and the function $K\times P \to G$, $(k,p)\mapsto kp$, is bijective. The Iwasawa decomposition shows that we can ...
Yemon Choi's user avatar
  • 25.5k
9 votes
1 answer
460 views

Does every non-amenable group contain a 2-generated non-amenable subgroup?

It is known that there are non-amenable groups not containing $F_2$, the free group on two generators; for example, Olshanskii's group. But does every non-amenable group contain a 2-generated non-...
user95282's user avatar
  • 997
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 ...
user33576's user avatar
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 ...
SMS's user avatar
  • 1,293
9 votes
1 answer
329 views

Can the Cesaro limit of a positive definite function be negative?

Let $G$ be a countable amenable group and $\gamma:G\to\mathbb{C}$ a positive (semi)definite function (i.e. such that $\gamma(g^{-1})=\overline{\gamma(g)}$ and $$\sum_{g,h\in G}f(g)\overline{f(h)}\...
Joel Moreira's user avatar
  • 1,701
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 ...
Ruy's user avatar
  • 2,233
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 ...
Alexander Pruss's user avatar
9 votes
0 answers
314 views

Is there an countable amenable dense subgroup of $U(\ell^2 \mathbb N)$?

Question: Does the unitary group $U(\ell^2 \mathbb N)$, equipped with the strong operator topology, contain a countable dense subgroup which is amenable as a discrete group? I would be also ...
Andreas Thom's user avatar
  • 25.3k
9 votes
0 answers
473 views

The approximation property of group C*-algebras

Let $G$ be a discrete group. Then the group C*-algebra $C^*(G)$ is nuclear if and only if $G$ is amenable. I am wondering whether nuclearity of $C^*(G)$ can fail for a Banach-space reason, namely due ...
Tomasz Kania's user avatar
  • 11.3k
8 votes
2 answers
426 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
8 votes
1 answer
284 views

Amenable inverse limits of torsionfree amenable groups

Let $$ \cdots \to \Gamma_n \to \Gamma_{n-1} \to \cdots \to \Gamma_0$$ be an inverse system countable groups and let's assume (for this post) that all homomorphisms in such an inverse system are ...
Andreas Thom's user avatar
  • 25.3k
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 ...
Diego Martinez's user avatar
8 votes
1 answer
569 views

Cake-cutting and amenable groups

I recently heard Alan Taylor speak about envy-free fair division and started wondering if questions like these make sense if we replace finitely additive measures with invariant means on amenable ...
Jon Bannon's user avatar
  • 6,977
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 \...
Narutaka OZAWA's user avatar
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 ...
Matt Zaremsky's user avatar
8 votes
0 answers
207 views

Superharmonic functions and amenability

Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$. Assume that there is a set of non-...
Kate Juschenko's user avatar
7 votes
3 answers
571 views

Finitely presented groups which are not residually amenable

What are examples of finitely presented but not residually amenable groups? Well, the examples that I want to have are simple f.p. groups as well as examples of non residually amenable groups arise ...
7 votes
2 answers
760 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
  • 189
7 votes
1 answer
290 views

Do syndetic sets on amenable semigroups have positive upper density?

Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $ a Folner sequence. For $S\subset \mathbb{G}$ define the upper density as $D^{\ast }(S)=\limsup_{n\rightarrow \infty }\...
Rob's user avatar
  • 123
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} \...
Adam's user avatar
  • 323
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 ...
ARG's user avatar
  • 4,342