# Questions tagged [amenability]

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### 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 $A$ ...
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### 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 ...
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### 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 (...
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### 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 \...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
$\newcommand{\Cstar}{C^*_{\text{red}}}\newcommand{\G}{\mathscr G}\newcommand{\H}{\mathscr H}$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 148 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 ... 7 votes 1 answer 284 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 391 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, ... 0 votes 1 answer 82 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 ... 11 votes 1 answer 311 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 ... 0 votes 2 answers 188 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... 6 votes 2 answers 655 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 61 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 ... 1 vote 1 answer 140 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 ... 2 votes 0 answers 130 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 ... 4 votes 1 answer 228 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 ... 5 votes 2 answers 328 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 ... 3 votes 2 answers 215 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 ... 6 votes 1 answer 205 views ### Introductory text on amenability I am looking for a book that covers amenability rigorously. Preferably a book aimed at beginners. 2 votes 0 answers 63 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 ... 8 votes 2 answers 409 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 ... 3 votes 1 answer 401 views ### Is a product of Følner sets Følner? Let G be an amenable (countable, discrete) group and let F_1,F_2,...,F_n,... and G_1,G_2,...,G_n,... be two Følner sequences. Is the product sequence (i.e. the sequence (H_n) where H_n is ... 1 vote 1 answer 154 views ### A question about Johnson's theorem on the first and second cohomology of commutative amenable algebras Johnson in cohomology of Banach algebra proved the following proposition. I need to some guidance for the bold part of the following proof. Do you know any papers or book with more details for this ... 4 votes 1 answer 364 views ### First and second cohomology groups of Banach algebras Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra A, one has H^1(A,X)=H^2(A,X)=0, ... 2 votes 0 answers 45 views ### Weak amenability hereditary properties Let \mathcal{A} be a commutative weakly amenable Banach algebra and \mathcal{B} be a Banach algebra, let \theta:\mathcal{A} \to \mathcal{B} be a continuous homomorphism with dense range; then it ... 4 votes 0 answers 128 views ### Contractible Banach algebras A Banach algebra A is contractible if H^1(A, X)=0 for all Banach A-bimodules X. Now to my question Let A be Banach algebra and I be closed ideal of A. If I and A/I are both ... 0 votes 0 answers 39 views ### If P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\} Prove \overline{\widehat{P(G)}}^* = \mathfrak{M}(G) Let \mathfrak{M}(G) be the set of all means on L_\infty (G) If P(G)=\{f\in L_1(G): f\geq 0, \int f d\lambda(x)=1\} Prove \overline{\widehat{P(G)}}^* = \mathfrak{M}(G) My attempt: We know ... 12 votes 1 answer 363 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 ... 5 votes 1 answer 245 views ### example of a non-amenable l.c. group such that C_r^*(G) satisfies the UCT Are there known any examples of non-amenable locally compact (or more restrictive, non-amenable discrete) groups G for which the reduced group C^*-algebra C_r^*(G) satisfies the universal ... 2 votes 0 answers 156 views ### Fixed-point properties for affine actions of topological groups T. Mitchell [Illinois J. Math. 14 (1970) 630--641] defined four properties of a topological semigroup, and in particular of a topological group G. Two of them are: (F2) Every jointly continuous ... 4 votes 3 answers 449 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 ... 4 votes 1 answer 275 views ### A question about spectral properties of a non-amenable group Let G be a group generated by a,b (for the sake of simplicity). Consider the element$$S=a+b+a^{-1}+b^{-1}\in{\mathbb C}[G], which may also be interpreted as an operator in $l^2(G)$ (by left ...
I know Higman group has an amalgamated product decomposition of $BS(1, 2)$. Is it possible to decompose Higman group as some groups we are more familiar with. For example, is there a normal subgroup K ...