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 $Γ$ 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 $Γ$ fixes a vertex of this decomposition, and the finite subgroups in each bag have uniformly bounded order. Moreover, a finitely generated group $Γ$ 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 $Γ$-equivariantly isomorphic to the Bass--Serre tree arising from a splitting of $Γ$ as a finite graph of finite groups.

Figures (3)

• Figure 1: A finite portion of the Bass--Serre decomposition tree for $\mathrm{SL}(2,\mathbb{Z}) \cong C_6 *_ {C_2} C_4$.
• Figure 2: Schematic Cayley graph (left) and decomposition tree (right).
• Figure 3: Conceptual sketch of three geometric obstacles preventing groups from satisfying Theorem \ref{['thm:char-torsion-free-simple']}

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof