Questions tagged [amenability]
The amenability tag has no usage guidance.
18
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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 $G$ ...
- 9,631
64
votes
7
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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 ...
- 2,762
2
votes
1
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${\rm II}_1$-factors with finite commutant: $\mathcal{A} \cap \mathcal{B} = \mathbb{C} \Rightarrow \mathcal{A}' \cap \mathcal{B}'$ hyperfinite?
Let $\mathcal{A} , \mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}', \mathcal{B}' $ are also a ${\rm II}_1$-factors.
Question: $\mathcal{A} \cap \mathcal{B} = \mathbb{...
- 25.9k
15
votes
1
answer
979
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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 ...
- 25.3k
12
votes
0
answers
271
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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 $...
- 4,342
12
votes
1
answer
478
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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 ...
- 121
11
votes
1
answer
559
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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 ...
- 213
10
votes
3
answers
849
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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 ...
- 723
9
votes
1
answer
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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-...
- 997
7
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2
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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 ...
- 4,342
6
votes
1
answer
293
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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 ...
- 9,631
6
votes
0
answers
446
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Does the Approximation Property (AP) pass to quotients by amenable subgroups?
Given a countable group $G$ with the AP, and an normal, amenable subgroup $N$ of $G$, does $G/N$ have the AP?
In particular, does there exist a group $G$ with the AP and a surjective group ...
- 3,315
6
votes
1
answer
1k
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a paradoxical decomposition of a group
Just my curiosity... Are there proofs the following fact, which does not involve Hall's matching theorem:
A group $\Gamma$ is amenable if and only if it does not admit a paradoxical decomposition.
...
- 4,695
5
votes
1
answer
298
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Amenability of $S^{\infty}$
Let $G$ be the group of all permutations of $\mathbb{N}$. If I am not mistaken, this group is denoted by $S^{\infty}$.
Is there a precise locally compact topology on $G$ such that $G$ would ...
- 312
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0
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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 ...
- 4,342
3
votes
1
answer
218
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Quotient of weak amenable Banach algebras
Let $A$ be a weak amenable Banach algebra and $I$ be a closed (two-sided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach ...
- 565
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votes
1
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261
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Two kinds of invariance of full conditional probabilities
Given a field $F$ of subsets of $\Omega$, we can define full conditional probabilities to be a function $P:F\times (F-\{ \varnothing \}) \to [0,1]$ such that:
$P(-|B)$ is a finitely-additive ...
- 2,383
2
votes
2
answers
518
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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 ...