Coarse separation and splittings in hyperbolic groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917351816953856 |
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| author | Bensaid, Oussama Genevois, Anthony Tessera, Romain |
| author_facet | Bensaid, Oussama Genevois, Anthony Tessera, Romain |
| contents | We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_17852 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coarse separation and splittings in hyperbolic groups Bensaid, Oussama Genevois, Anthony Tessera, Romain Group Theory Geometric Topology Metric Geometry 20F65, 20F69 We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth. |
| title | Coarse separation and splittings in hyperbolic groups |
| topic | Group Theory Geometric Topology Metric Geometry 20F65, 20F69 |
| url | https://arxiv.org/abs/2603.17852 |