Virtual homological torsion in graphs of free groups with cyclic edge groups

TL;DR

This paper proves that in certain hyperbolic groups formed by graphs of free groups with cyclic edge groups, all finite abelian groups appear in the abelianization of some finite-index subgroup, with implications for profinite rigidity.

Contribution

It establishes the universality of finite abelian groups in the abelianizations of finite-index subgroups for a broad class of hyperbolic groups, except for specific free product cases.

Findings

Finite abelian groups appear as direct summands in abelianizations of finite-index subgroups.

Free products of free and surface groups are profinitely rigid among these hyperbolic groups.

Partial surface words are uniquely determined by their induced word measures on finite groups.

Abstract

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in the abelianization of some finite-index subgroup $G'\le G$. As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.