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Every nontrivial group $G$ satisfies your condition (iii). To see why, note that for any subgroup $H$ of $G$, the left action of $H$ on $G$ is free. So if $X$ is a set of representatives of the orbits …

Here is an elementary way to prove that there can´t be a finite group $G$ and non-identity $g\in G$ with $r_2(g)>\frac{5}{6}|G|$. Suppose that happens and call $S=\{x\in G; x^2=g\}$. Then of course th …

By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence. Context: I just came up with this question and surprisingly I haven' …

I found an answer to Question 1 little after asking it, but as it was part of the material I was planning to upload to arXiv (and since the proof is long so I prefer to just cite it), I decided to wai …

Let $G$ be a countable amenable group. We consider sequences $(z_g)_{g\in G}$ of complex numbers with $|z_g|=1$ for all $g\in G$. I will say $(z_g)_{g\in G}$ is background noise for a (left-)Følner se …

This is crossposted from MSE. We say a subset $A$ of a group $G$ is a monotile for $G$ if $G$ is a disjoint union of right translates of $A$. In his article Monotileable Amenable Groups, B. Weiss give …

$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE. Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq …