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6
votes
1answer
159 views

Free actions of non-amenable groups

Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free ...
2
votes
0answers
68 views

Non-ergodic Dye Theorem for orbit equivalent automorphisms

The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent. Question: Is there a version of the above theorem for non-ergodic ...
8
votes
1answer
118 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 ...
6
votes
0answers
286 views

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
votes
1answer
192 views

Can the full and reduced group $C^*$-algebras be “noncanonically” isomorphic?

Is there a locally compact group $G$ such that the canonical map from $C^{*}(G)$ to $C^{*}_{red} G$ is not isomorphism, hence $G$ is not amenable but these two $C^{*}$ algebras are isomorphic ...
4
votes
1answer
230 views

Characterizing residually amenable groups

Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as ...
16
votes
1answer
757 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, ...
4
votes
0answers
88 views

Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...
4
votes
1answer
210 views

When does convolution preserve the `size' of a function?

For a positive function $f$ and positive measures $\mu, \nu$, does $$\mu\ast f\leq \nu\ast f \Rightarrow \|\mu\|\leq \|\nu\|?$$ More details: Let $G$ be a locally compact group, $C(G)$ be the space ...
1
vote
1answer
161 views

Infinite amenable group subfactors

Let amenable groups $\Gamma$ and $\Gamma'$. They act outerly of only one manner on the hyperfinite ${\rm II}_1$-factor $\mathcal{R}$. Question: $(\mathcal{R} \subset \mathcal{R} \rtimes \Gamma) ...
2
votes
1answer
190 views

${\rm II}_1$-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} = ...
11
votes
0answers
214 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 ...
9
votes
1answer
201 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 ...
3
votes
1answer
168 views

Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...
5
votes
0answers
193 views

Is translation by the free group (in two generators) on a certain completion of the group an amenable action?

Let $\mathbb{F}_2 = \langle a,b\rangle$ be the free group in two generators $a,b$ and let $\alpha \in \text{End}(\mathbb{F}_2)$ be given by $\alpha(a) = a^2, \alpha(b)= b^2$. Note that the index ...
0
votes
1answer
127 views

Amenability of the Koopman representation

Let $G$ be a locally compact group which acts non-singularly on a standard probability space $(X,\mu)$. Consider the Koopman representation $\pi_X:G\rightarrow U(L^2(X,\mu))$ defined by ...
8
votes
0answers
306 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 ...
1
vote
2answers
270 views

Amenability of $l^\infty$ [closed]

I'm working on the amenability of some Banach algebras, and I'm wondering why $l^\infty$ is amenable ? Does any one has any idea how to start ? Thank you in advance.
3
votes
0answers
114 views

Pointwise ergodic theorem for amenable semigroups

Using tempered Folner sequences one may show a pointwise ergodic theorem for amenable groups. (see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full) Is there a similar ...
6
votes
1answer
169 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 ...
2
votes
1answer
193 views

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 ...
4
votes
0answers
71 views

Estimates for simple random walks in groups of intermediate growth

I'm looking for references for the rate of escape and return probability for a group of intermediate growth. Let $0<\alpha < 1$. If the volume growth is $\succeq \mathrm{exp}(n^\alpha)$, then ...
2
votes
1answer
160 views

Relative amenability of subgroups

Let $\Gamma$ be a countable group and let $\Lambda_1,\Lambda_2<\Gamma$ be subgroups. We say that $\Lambda_1$ is amenable relative to $\Lambda_2$ if the action of $\Lambda_1$ on $\Gamma/\Lambda_2$ ...
10
votes
0answers
141 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 ...
0
votes
0answers
103 views

Amenability of the pair $(GL(2,\mathbb{Q})^+,SL(2,\mathbb{Z}))$

I am trying to see whether the pair $(GL(2,\mathbb{Q})^+,SL(2,\mathbb{Z}))$ is amenable in the following sense: Let $H$ be a closed subgroup of a locally compact group $G$. The pair $(G,H)$ is called ...
9
votes
0answers
172 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 ...
2
votes
1answer
196 views

Does supramenability imply that $a+c=b+2c \Rightarrow a=b+c$ on the type semigroup?

Tarski proved that if a group $G$ is exponentially bounded, then for $a$, $b$ and $c$ in the associated (equidecomposability) type semigroup, we have $a+c=b+2c \Rightarrow a=b+c$. Question: Can ...
3
votes
1answer
206 views

A stronger version of supramenability?

A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
8
votes
0answers
369 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 ...
54
votes
1answer
3k 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 $G$ ...
1
vote
1answer
117 views

Amenable normal closure

Prove or disprove: Let $G$ be a countable group. Let $H < G$ be an amenable subgroup with a finite conjugacy class. Then the normal closure of $H$ is also amenable. Thanks!
6
votes
1answer
781 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 ...
4
votes
1answer
362 views

Probabilities of a random walk exiting a set

Let $F$ be a finite connected set in a graph (soon to be the Cayley graph of a group) and $\mathrm{Ex}_x^F$ be the function on the vertices in $F^c$ which are neighbour to vertices in $F$ defined as ...
3
votes
0answers
100 views

Cogrowth and value of its series at the critical exponent

Let $G$ be a finitely generated group and write $G = F/N$ for $N$ a normal subgroup of a free group $F$. Let $S_n$ be the elements in $F$ written as words of exactly $n$ letters. So, for $n\geq 1$, ...
3
votes
1answer
184 views

Spectral synthesis for central functions on locally compact groups

There is a large literature on harmonic analysis on locally compact group, that I am just beginning to discover. However I have not seen so far anything that emphasizes the central functions on $G$. A ...
16
votes
3answers
451 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 ...
3
votes
1answer
395 views

Existence of nice Folner sequences

I'm attempting a proof by induction and, for the inductive step, it would be very useful for me to have some control on a Folner sequence. Indeed, let $G$ be a finitely generated amenable group, fix a ...
5
votes
1answer
213 views

Cocycles for right- and left- regular representations on $\ell_2(G)$

Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on ...
3
votes
2answers
556 views

Another question about amenability and Følner sequences

Følner's characterization of Amenability says that a group $G$ is amenable if there exists a directed set $(I,\leq)$ and a net {$F_i:i\in I$} of finite subsets of $G$ such that for every $γ ∈ G$, ...
4
votes
1answer
263 views

left- and right- Folner sets

Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m ...
5
votes
0answers
298 views

The kernel of all invariant means

Let $G$ be discrete group which is amenable (i.e. it admits an left-invariant mean, i.e. a continuous positive normalised linear functional on $m:\ell^\infty(G) \to \mathbb{R}$ such that $\forall g ...
5
votes
0answers
206 views

Can invariant means be really considered as the generalization of the uniform measure?

I am writing a paper for game theorists where I use (countable) amenable groups to do some things. So I am writing up a preliminary section about countable amenable groups whose main purpose is to ...
10
votes
1answer
312 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
0answers
445 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 $xy\in W$. ...
7
votes
1answer
441 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 ...
5
votes
0answers
201 views

Can a non-amenable group have a 'centrally invariant mean'?

Let $G$ be a countable, discrete group, and $f\in\ell^\infty(G)$. Let me say that $f$ has a centrally invariant mean if there is a finitely additive probability measure $\mu$ on $G\times G$ such that ...
1
vote
0answers
443 views

Why groups that admit Folner Sequences are amenable

I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...
1
vote
1answer
191 views

Left mean values vs right mean values

Let $G$ be a countable amenable group and $f\in\ell^\infty(G)$. Denote by $L,R,I$ respetively the sets of left-, right- and bi-invariant means on $G$. Denote by $M_L(f)$ (resp. $M_R(f),M_I(f)$) be the ...
6
votes
1answer
401 views

A variational principle for amenable groups

Update: If somebody is interested, in Sec. 3, Theorem 3.5, of http://arxiv.org/abs/1203.2301 the variational principle for amenable groups such that every conjugacy class is finite is proved. Let $G$ ...
3
votes
1answer
333 views

Fubini's theorem and unique mean value

Following the terminology of Rosenblatt, I will say that a bounded function $f:\mathbb Z\rightarrow\mathbb R$ has a unique mean value if for every pair of finitely additive translation invariant ...