All Questions
Tagged with amenability measure-theory
7 questions
12
votes
2
answers
362
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 ...
- 11.3k
9
votes
1
answer
773
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 ...
- 2,555
5
votes
1
answer
630
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 well?...
- 11.3k
3
votes
1
answer
401
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 ...
- 6,034
3
votes
1
answer
298
views
Averaging measurable functions over amenable group actions
Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the ...
- 10.5k
0
votes
2
answers
224
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 $|Gx| = n$...
- 19.7k
-1
votes
1
answer
168
views
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
\...
- 741