Questions tagged [amenability]
The amenability tag has no usage guidance.
160
questions
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 $G$ ...
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 ...
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 ...
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)||\...
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 $xy\in W$. ...
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 ...
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, ...
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 ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $...
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 ...
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 (...
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 ...
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 ...
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 ...
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 ...
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 (...
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 ...
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-...
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 ...
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
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)}\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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
$$\max_{s\in E} |s F \mathbin{\triangle} F| \le \...
8
votes
0
answers
148
views
Amenable automatic groups
Are there any known examples of finitely generated groups that are both amenable and automatic, besides the easy example of virtually abelian groups? Or are there any known restrictions that arise if ...
8
votes
0
answers
207
views
Superharmonic functions and amenability
Let $G$ be a group generated by a finite set $S$. Let $P$ be a Markov operator defined by the uniform measure on $S$. A function is superharmonic if $Pf\leq f$.
Assume that there is a set of non-...
7
votes
3
answers
571
views
Finitely presented groups which are not residually amenable
What are examples of finitely presented but not residually amenable groups?
Well, the examples that I want to have are simple f.p. groups as well as examples of non residually amenable groups arise ...
7
votes
2
answers
760
views
Amenable action intuition
Let $\Gamma$ be a discrete group and $A$ be a $C^*$-algebra. Consider an action $\alpha: \Gamma \to \operatorname{Aut}(A)$. There is a notion of amenability for such an action (see e.g. Brown and ...
7
votes
1
answer
290
views
Do syndetic sets on amenable semigroups have positive upper density?
Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $
a Folner sequence.
For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\...
7
votes
1
answer
153
views
Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
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 ...