8 problems

Let $G$ and $H$ be finite groups, and let $G\wr\mathbb{Z}$ and $H\wr\mathbb{Z}$ be their lamplighter groups. Conjecture I. Then $G\wr\mathbb{Z}$ is quasi-isometric to…

Let $G$ be a nontrivial finite group, let $G\wr\mathbb{Z}=(\bigoplus_{\mathbb{Z}}G)\rtimes\mathbb{Z}$ be its lamplighter group, and let $G((t))$ denote the associated boundary spac…

Let $\mathbb{F}_q((t))$ be the Laurent-series field over the finite field $\mathbb{F}_q$, and let $\operatorname{Bilip}(\mathbb{F}_q((t)))$ denote its bilipschitz self-homeomorphis…

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Consider the $\lambda$-stationary random walk on the lamplighter group $H\wr G$. Critical recurr…

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Let $\lambda_c$ be the critical parameter such that the $\lambda$-stationary random walk on…

The lamplighter group's non-semistability conjecture. The group $L$ is not semistable at infinity.

Let ${\mathbb Z}^2 \wr {\mathbb Z}_2$ be the lamplighter group on the plane, equipped with the usual switch-walk-switch generating set. Connectedness conjecture. Every sphere in it…

Let $\mathcal{T}$ be an oriented tree with a fixed end, and let $(Z_n)$ be a random walk with bounded range on $\mathbb{Z}_2\wr\mathcal{T}$. Write $(X_n)$ for its projection to…