Rigidity and flexibility results for groups with a common cocompact envelope

Abstract

A locally compact group $G$ is a cocompact envelope of a group $Γ$ if $G$ contains a copy of $Γ$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $Γ,Λ$ having a common cocompact envelope, and asks what properties must be shared between $Γ$ and $Λ$. We first consider the setting where the common cocompact envelope is totally disconnected. In that situation we show that if $Γ$ admits a finitely generated nilpotent normal subgroup $A$, then virtually $Λ$ admits a normal subgroup $B$ such that $A$ and $B$ are virtually isomorphic. We establish both rigidity and flexibility results when $Γ$ belongs to the class of solvable groups of finite rank. On the rigidity perspective, we show that if $Γ$ is solvable of finite rank, and the locally finite radical of $Λ$ is finite, then $Λ$ must be virtually solvable of finite rank. On the flexibility perspective, we exhibit groups $Γ,Λ$ with a common cocompact envelope such that $Γ$ is solvable of finite rank, while $Λ$ is not virtually solvable. In particular the class of solvable groups of finite rank is not QI-rigid. We also exhibit flexibility behaviours among finitely presented groups, and more generally among groups with type $F_n$ for arbitrary $n \geq 1$.