Questions tagged [amenability]

Use this tag for questions related to amenable groups, which are locally compact topological groups carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements.

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Number of maximal subgroups in finitely generated amenable groups

The following statement is known to be true: Any subgroup of a finitely generated group lies in a maximal subgroup Proof: Suppose, $G = \langle \{x_1, … , x_n\} \rangle$ is a counterexample. Then ...
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490 views

Could we define an average that satisfied all the conditions below? If so, how?

I posted this question on Math Overflow if you wish to answer there. Consider $P:A\to\mathbb{R}$, where $A$ is a subset of $\mathbb{R}$. I want the average of $P$ to ALWAYS be between the infimum ...
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1answer
20 views

$S-S$ is syndetic set if $S$ has positive upper density, in the case of group action

Let $G$ be a discrete group. A sequence $\mathcal{F}=\{F_n\}$ is called a $Folner$ sequence if $\frac{|gF_n\Delta F_n|}{|F_n|}\to 0$ as $n\to \infty$ for every $g\in G$. $F_n$ is a finite subset of $...
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Help with definition of amenable groups.

I am reading some lecture notes that have the following definition of a left invariant mean on a group: $G$ is called amenable if there exists a bounded linear functional $m:L^{\infty}(G) \to \mathbb{...
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$\{T_n\}$ Folner $\implies \{S_n\} = \{\bigcup_{k=1}^{n}T_k\}$ Folner?

Given an countable amenable group $G$, let $\{T_n\}_{n \in \mathbb{N}}$ be a Folner sequence for $G$, i.e., $\lim_{n \to +\infty} \frac{|gT_n \Delta T_n|}{|T_n|} = 0$, for every $g \in G$. Now, for ...
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1answer
99 views

How to prove that the Reiter property implies the Folner property?

Let $G$ be a group acting on a set $X$ such that the Reiter property holds, i.e. $$ \forall S\subset G \text{ finite, }\forall \varepsilon>0,\ \exists\ \varphi\in\mathcal{l}^1(X),\ \forall s\in S: \...
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1answer
27 views

How to find an example of $G$ amenable such that $G^\mathbb{N}$ isn't amenable?

What is an example of an amenable group $G$ such that $G^\mathbb{N}$ isn't amenable? My thoughts: Consider for example $G=\{\begin{pmatrix}a&b\\0&1\end{pmatrix}\ |\ a,b\in\mathbb{R}, a\neq 0\}...
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16 views

Unsure about the definition of a mean on a locally compact group.

I am reading Lectures on Amenability, and I am a little unsure about the definition of a mean: Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}(G)$ containing the constant ...
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Example of Modified Folner Net

Suppose $A_i$ is a subset of $A$ and $A$ is a subset of $\mathbb{R}$ Let $A$ be a semigroup and let $\mathcal{F}=\langle F_{n} \rangle_{n\in D}$ be a net in $\mathcal{P}_{f}(A)$, where $\mathcal{...
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Amenable Group and Bounded Cohomology

A group $G$ is said to be amenable if for every bounded function $f:G \to \mathbb{R}$, there exists a mean $I(f) \in \mathbb{R}$ such that $I(f)\geq 0$ whenever $f \geq 0$, $$I(\mathbb{1}) =1$$ and is ...
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631 views

How to rigorously define a translation-invariant measure that follows these requirements?

I am unsure anyone at math stack exchange can answer my question, so I moved it to MathOverflow. Let $A$ be a subset of $\mathbb{R}$. I want to rigorously define what I believe is the most "intuitive ...
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180 views

How to use Programming to Approximate the Density of an arbitrary set which uses the Folner Sequence of it's superset?

**EDIT: I followed @ZacharyHunter's advice in the comments. I am unable to correctly calculate Exists in my random sample. EDIT: I changed my code. Everything ...
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1answer
88 views

Amenable groups: From fixed point property to invariant mean.

Definitions Let $G$ be a Lie group and $\mu$ be a Haar measure on $G$. A compact-convex $G$-space is a $G$-invariant compact and convex subset of a locally convex topological vector space $V$ on ...
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1answer
215 views

How to calculate the density of any subset of $\mathbb{Q}$ using a particular Folner Sequence of $\mathbb{Q}$?

Suppose we choose a particular Folner Sequence of $\mathbb{Q}$ such as $F_n=\left\{\frac{p}{2^k(2q+1)}:p,q\in\mathbb{Z},k\in\mathbb{N},\gcd\left(p,2^k(2q+1) \right)=1,2^k\le n, |2q+1| \le n, \left|\...
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1answer
181 views

Could a “Modified” Density of subsets of $[a,b]$, using a “modified” Folner Net, give the same results as the Lebesgue measure of these subsets?

I read a research paper stating we can extend the definition of a Density, that uses a Folner Sequence of countable sets, to one that uses Folner Nets of Uncountable sets (ex: $\mathbb{R}$). I couldn’...
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42 views

Infinite finitely generated amenable periodic groups

I know that the Grigorchuk group is an example of this. I also know that there are other Grigorchuk groups that satisfy this as well. Are there any other examples? Is any general structure/...
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20 views

Could we reduce the number of Folner Sequences used for the density of the subset of Rational Numbers?

According to the comments in this answer, there are multiple Folner Sequences $F_n$ of $\mathbb{Q}$, one can use for the density of $T$, a subset of rational numbers. $$D(T)=\lim_{n\to\infty}\frac{|T\...
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278 views

Can an integral defined by a density with respect to folner sequences of $A$ exist on a function defined on subsets of $A$?

Consider the following function where $A=\mathbb{Q}$ $$F(x)=\begin{cases} 2^x & x=A_1\\ x^2 & x=A_2\\ \text{Undefined} & \text{Everywhere Else} \end{cases} $$ $$A_1=\left\{\frac{2m+1}{2n+...
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38 views

Amenability discrete groups vs. locally compact groups

I´ve been reading about amenability of groups, and I don´t know if the definition involving the existence of $\mu$ a finitely additive, left-invariant mean on $\mathbb{B}(G)$(the Borel set on G) with $...
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17 views

In amenable Banach algebra if Jacobson radical be finite dimensional then must be semisimple

Let $\mathcal{A}$ be an amenable Banach algebra, I'd like to prove if Jacobson radical rad$\mathcal{A}$ be finite dimensional, then rad$\mathcal{A}$ must be semisimple I know in an amenable ...
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35 views

Day’s Theorem for a locally compact group

The result below is originally due to Day when $H$ is a locally compact group Theorm Let $H$ be a locally compact group, $H$ is amenable if and only if there is a net $\{f_\alpha\}_\alpha \subset L^...
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31 views

References books and lecture notes for Amenability

I am reading the book "Lectures on amenability" by Volker Runde I was wondering if someone could suggest me some books and Lecture notes with some good problems to go over My backgrounds are the ...
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1answer
57 views

Equivalence on definition of Folner sequence

$\textbf{Definition 1.}$ Let $\{F_n\}_{n \in \mathbb{N}}$ be a sequence of finite subsets of $\mathbb{Z}^d$. We say that $\{F_n\}_{n \in \mathbb{N}}$ is a Folner sequence if for every $v \in \mathbb{Z}...
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2answers
93 views

Index of subgroups in residually finite groups

Let $G$ be an infinite finitely generated residually finite group. Is it true that $G$ contains finite-index subgroups of arbitrarily large index? What about the converse: does there exist a finitely ...
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1answer
35 views

Strong Folner condition(SFC) implies the existence of a left Følner sequence.

I got stuck with this problem while reading Density in Arbitrary Semigroups by Hindman and Strauss. It says: Problem: If $S$ is a countable semigroup. Then SFC on $S$ implies the existence of a left ...
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1answer
185 views

Extensions of amenable groups

Let $1 \to N \to G \to Q \to 1$ be an extension of (discrete) groups, where $N$ and $Q$ are amenable. Using the fixed-point theorem, I know how to show that $G$ is amenable. However, I was wondering ...
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26 views

Bound on the measure of a sum of sets

Let $G$ be a locally compact abelian group (I am mostly interested in $\mathbb{R}^d$) with Haar measure $\mu$ be the Haar measure. Consider a fixed compact set $K$ containing the identity. Is it true ...
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1answer
62 views

Amenability of countable product of a finite group.

Given a finite group $G$, I wish to show that the that product $\prod_{n \in \mathbb{N}} G$ is an amenable group. It is known that for a group $H$, $H$ is amenable if and only if for all finite ...
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41 views

HNN embedding theorem for countable groups not preserving amenability

The well-known embedding theorem for countable groups by Higman-Neumann-Neumann says that Theorem 1: Every countable group can be embedded in a 2-generated group. One can prove this theorem using ...
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119 views

Show that free group $\mathbb{F}_2$ is not amenable using Hulanicki-Reiter condition.

Hulanicki-Reiter condition states: A finetely generated group G is amenable if and only if for every $\epsilon>0$ and $R>0$ there exists a function $f\in l_1(G)_{1,+}$ such that: a) $||f-\...
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2answers
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The positive net of a weak* convergent net is weak* convergent.

Suppose $X$ is a unital $C^*$-algebra, $i:X\to X^{**}$ is the natural isometric inclusion as Banach space. Denote $S_X=\{x\in X:\|x\|=1\}$, $S_{X,+}=\{x\in X:\|x\|=1,x\ge 0\}$, so does $S_{X^{**},+}$. ...
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1answer
128 views

Polynomially sized Folner sets

Let $\Gamma$ be a finitely-generated group with a fixed finite generating set $S$. Then, $\Gamma$ is amenable if and only if it there is a sequence $(F_n)_{n=1}^{\infty}$ of finite subsets of $\Gamma$...
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1answer
32 views

existence of weak* continuous means for amenable locally compact groups

Let G be an amenable locally compact group. Does there exist a left invariant mean $m \colon L^\infty(G) \to \mathbb{C}$ on $L^\infty(G)$ which is in addition weak*-continuous ? Recall that the von ...
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1answer
69 views

Amenability of a directed union

I have the following statement: if all groups in the directed system $\{G_{i}\}_{i\in I}$ are amenable, then so is their directed union $G:= \bigcup_{i\in I}G_{i}$ (Remember that a group is amenable ...
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1answer
131 views

Are there discrete uncountable groups with Folner sequences?

For discrete groups, the Folner characterization of amenability says that for any finite subset $K\subseteq G$, and any $\varepsilon$, there exists a finite subset $F$ such that $|F\Delta KF|<\...
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142 views

Is SO3(C) an amenable group?

It is well known that SO(3) is an example of non-amenable group, but it is not so easy to find information about the amenability of its complex counterpart.
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56 views

Use of the condition "compact' in a proof

Let $\Gamma$ be an abelian amenable group acting on a compact space $X$. Show that there is an invariant Radon measure $\mu$ on $X$. Proof: Fix $x_0 \in X$. For $f \in C(X)$ define $g_f(t)=f(t.x_0)$....
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1answer
261 views

Is there a connection between my density formula and an invariant mean defined by a folner sequence of rational numbers?

I am a first-year undergraduate who stumbled upon natural density. I am working on extending this definition to the subset of rational numbers. While most people would wait until they are older, I am ...
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92 views

Amenable groups and action

It is known that if a group $G$ acts amenably on a set $X$ and the stabilizer of every point in $X$ is amenable, then $G$ itself is amenable. Is there any elementary proof of this fact? Thanks!
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Why can't a non-amenable Torsion group have a non-abelian free subgroup

I'm trying to work through some proofs in and article which is talking about Tarski numbers. There is a proposition which says that For a group G one has T(G)=4 if and only if G contains a non-abelian ...
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1answer
97 views

State of $l^{\infty}(G)$

Recently I've come across the definition of an amenable group, which says the following: "A countable discrete group $G$ is amenable if there exist a state $\mu$ on $l ^{\infty}(G)$ which is ...
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95 views

Applications of amenable groups

I will do a talk about amenablity for a public of master students who are mainly interested in algebraic geometry and mathematical physics. At the end, I would like to include some links with some ...
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1answer
142 views

Amenable group $G$ and a (left) invariant mean on it.

While reading "The Banach-Tarski Paradox" by S. Wagon, I faced the definition of a amenable group: a group that possesses a finitely additive, $G$-invariant measure on all subsets, with total measure ...
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114 views

Pointwise ergodic theorem & amenable semigroups

Lindenstrauss proved pointwise ergodic theorem for amenable groups using tempered folner sequence. Is there a pointwise ergodic theorem for amenable semigroups such as $\mathbb{N}^2$?
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1answer
222 views

quotients of amenable groups are amenable

I want to proof that for a discrete group $G$ and $H$ a normal subgroup of $G$ the following holds: $G$ and $H$ are amenable $\Rightarrow$ the quotient $G/H$ is amenable (using one of the ...
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1answer
235 views

Amenability of the integers (with the Følner condition)

I only know the following definition of amenability of a discrete group (but I think the definition is suitable for non-dicrete groups as well): Def: A discrete group $G$ satisfies the Følner ...
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1answer
112 views

Example of non-amenable group which is the inverse limit of amenable groups

1) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$? 2) Does there exist a non-amenable locally compact group $G$ which ...
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182 views

Day's fixed point theorem

Day's fixed point theorem (Theorem 1.3.1; Lecture on amenability; Volker Runde) Let $G$ be a locally compact group. The following are equivalent: $G$ is amenable. If $G$ acts (from left side) ...
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244 views

Is there an example of a non compact, semisimple, amenable Lie group?

By semisimple I mean the real Lie algebra of $G$ is semisimple. I guess there is not but I can't formulate a rigorous argument.
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1answer
44 views

Number of copies of irreducible unitary representation in $L^2(G)$ for compact group $G$?

Peter-Weyl Theorem is concerned with expressing $L^2(G)$ as closure of direct sum of subspace generated by irreducible unitary representation. Every irreducible representation on a compact group is ...