Rigidity and flexibility results for groups with a common cocompact envelope

A locally compact group $G$ is a cocompact envelope of a group $\Gamma$ if $G$ contains a copy of $\Gamma$ as a discrete and cocompact subgroup. We study the problem that takes two finitely generated groups $\Gamma,\Lambda$ having a common cocompact envelope, and asks what properties must be shared between $\Gamma$ and $\Lambda$. We first consider the setting where the common cocompact envelope is totally disconnected. In that situation we show that if $\Gamma$ admits a finitely generated nilpotent normal subgroup $A$, then virtually $\Lambda$ admits a normal subgroup $B$ such that $A$ and $B$ are virtually isomorphic. We establish both rigidity and flexibility results when $\Gamma$ belongs to the class of solvable groups of finite rank. On the rigidity perspective, we show that if $\Gamma$ is solvable of finite rank, and the locally finite radical of $\Lambda$ is finite, then $\Lambda$ must be virtually solvable of finite rank. On the flexibility perspective, we exhibit groups $\Gamma,\Lambda$ with a common cocompact envelope such that $\Gamma$ is solvable of finite rank, while $\Lambda$ 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 $n \geq 1$.