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arXiv · 2505.20960

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

Dario Ascari·Jonathan Fruchter
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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.

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Dario Ascari, Jonathan Fruchter. 2025-05-27. Virtual homological torsion in graphs of free groups with cyclic edge groups. https://arxiv.org/abs/2505.20960

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