Rigidity and flexibility results for groups with a common cocompact envelope
Adrien Le Boudec
TL;DR
The paper investigates when two finitely generated groups $\Gamma$ and $\Lambda$ sharing a common cocompact envelope $G$ must reflect each other’s algebraic structure. It develops a robust framework for totally disconnected envelopes using Willis’ tidy subgroup theory to prove rigidity results, notably that a finitely generated nilpotent normal subgroup $A$ of $\Gamma$ with cobounded normalizer forces a corresponding virtually-isomorphic normal subgroup in $\Lambda$. It then analyzes solvable groups of finite rank, proving CE-rigidity under certain hypotheses while constructing explicit flexible counterexamples that show the absence of CE-rigidity (and even QI-rigidity) in broader settings, including finitely presented cases. The paper also addresses envelopes of polycyclic groups and lattices in Lie groups, showing that envelopes are typically compact-by-discrete and obtaining classifications in key polycyclic examples. Finally, it provides two flexible constructions via Diestel–Leader graphs and adelic-type envelopes demonstrating non-rigidity phenomena for solvable groups of finite rank, even among groups with strong finiteness properties, while also identifying CE-rigid subclasses at the $F_\infty$ end.
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 $n \geq 1$.