Virtual homological torsion in graphs of free groups with cyclic edge groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910970556710912 |
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| author | Ascari, Dario Fruchter, Jonathan |
| author_facet | Ascari, Dario Fruchter, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20960 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Virtual homological torsion in graphs of free groups with cyclic edge groups Ascari, Dario Fruchter, Jonathan Group Theory Geometric Topology 20F65, 20F67, 57M07 (Primary) 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. |
| title | Virtual homological torsion in graphs of free groups with cyclic edge groups |
| topic | Group Theory Geometric Topology 20F65, 20F67, 57M07 (Primary) |
| url | https://arxiv.org/abs/2505.20960 |