Finite Index Rigidity of Relatively Hyperbolic Groups
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908519147503616 |
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| author | Lazarovich, Nir Rahamim, Gon Sisto, Alessandro |
| author_facet | Lazarovich, Nir Rahamim, Gon Sisto, Alessandro |
| contents | We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume negatively curved manifolds, to limit groups, and to free-by-cyclic groups. More generally, we show that if two finite index subgroups of a relatively hyperbolic group are isomorphic via a map that respects their peripheral structures, then their indices in the ambient group are equal. The proof relies on demonstrating that the number of simplices in a simplicial classifying space of a finite index subgroup in a relatively hyperbolic group grows linearly with its index. These results generalize earlier work of the first author in the context of hyperbolic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite Index Rigidity of Relatively Hyperbolic Groups Lazarovich, Nir Rahamim, Gon Sisto, Alessandro Group Theory Geometric Topology 20F65, 20F67 We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume negatively curved manifolds, to limit groups, and to free-by-cyclic groups. More generally, we show that if two finite index subgroups of a relatively hyperbolic group are isomorphic via a map that respects their peripheral structures, then their indices in the ambient group are equal. The proof relies on demonstrating that the number of simplices in a simplicial classifying space of a finite index subgroup in a relatively hyperbolic group grows linearly with its index. These results generalize earlier work of the first author in the context of hyperbolic groups. |
| title | Finite Index Rigidity of Relatively Hyperbolic Groups |
| topic | Group Theory Geometric Topology 20F65, 20F67 |
| url | https://arxiv.org/abs/2509.04323 |