Combinatorial Characterizations of Virtually Torsion-Free and Virtually Free Groups

Abstract

We establish combinatorial characterizations of virtually torsion-free and virtually free groups using the canonical graph decomposition theory in \cite{DJKK22}. Our main results show that a finitely presented, residually finite group $\Gamma$ is virtually torsion-free if and only if there exists a locality parameter $r>0$ such that its $r$-local cover admits a canonical tree-decomposition with finite quotient and finite adhesion, every finite subgroup of $\Gamma$ fixes a vertex of this decomposition, and the finite subgroups in each bag have uniformly bounded order. Moreover, a finitely generated group $\Gamma$ is virtually free if and only if for some $r>0$ its $r$-global decomposition has a finite model graph with finite bags and the tree-decomposition of the $r$-local cover is $\Gamma$-equivariantly isomorphic to the Bass--Serre tree arising from a splitting of $\Gamma$ as a finite graph of finite groups.