Homological torsion growth in non-normal chains of graphs of free groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916956456615936 |
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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, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1 \ge G_2 \ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, $\liminf_{n\rightarrow \infty} \frac{\log \vert \mathrm{Tor}_p(G_n^{\mathrm{ab}})\vert}{[G:G_n]} >0$ (where $\mathrm{Tor}_p(G_n^{\mathrm{ab}}) = \{g \in G_n^{\mathrm{ab}} \;\vert\; g \text{ has order a power of } p\}$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_15075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homological torsion growth in non-normal chains of graphs of free groups Ascari, Dario Fruchter, Jonathan Group Theory Geometric Topology 20F65, 20F67, 57M07 Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups, and which is not isomorphic to a free product of free and surface groups. We show that $G$ admits an exhausting, nested sequence of finite-index non-normal subgroups $G\ge G_1 \ge G_2 \ge \cdots$ with exponential homological torsion growth. More specifically, we prove that simultaneously for every prime $p$, $\liminf_{n\rightarrow \infty} \frac{\log \vert \mathrm{Tor}_p(G_n^{\mathrm{ab}})\vert}{[G:G_n]} >0$ (where $\mathrm{Tor}_p(G_n^{\mathrm{ab}}) = \{g \in G_n^{\mathrm{ab}} \;\vert\; g \text{ has order a power of } p\}$). |
| title | Homological torsion growth in non-normal chains of graphs of free groups |
| topic | Group Theory Geometric Topology 20F65, 20F67, 57M07 |
| url | https://arxiv.org/abs/2509.15075 |