Frattini subgroups of hyperbolic-like groups
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917699923214336 |
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| author | Goffer, Gil Osin, Denis Rybak, Ekaterina |
| author_facet | Goffer, Gil Osin, Denis Rybak, Ekaterina |
| contents | We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group $G$ admitting a general type action on a hyperbolic space $S$, we show that the induced action of the Frattini subgroup $Φ(G)$ on $S$ has bounded orbits. This implies that $Φ(G)$ is "small" compared to $G$; in particular, $|G:Φ(G)|=\infty$. In contrast, for any finitely generated non-cyclic group $Q$ with $Φ(Q)=\{ 1\}$, we construct an infinite lacunary hyperbolic group $L$ such that $L/Φ(L)\cong Q$; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04592 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frattini subgroups of hyperbolic-like groups Goffer, Gil Osin, Denis Rybak, Ekaterina Group Theory 20F67, 20F65, 20F69 We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group $G$ admitting a general type action on a hyperbolic space $S$, we show that the induced action of the Frattini subgroup $Φ(G)$ on $S$ has bounded orbits. This implies that $Φ(G)$ is "small" compared to $G$; in particular, $|G:Φ(G)|=\infty$. In contrast, for any finitely generated non-cyclic group $Q$ with $Φ(Q)=\{ 1\}$, we construct an infinite lacunary hyperbolic group $L$ such that $L/Φ(L)\cong Q$; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups. |
| title | Frattini subgroups of hyperbolic-like groups |
| topic | Group Theory 20F67, 20F65, 20F69 |
| url | https://arxiv.org/abs/2402.04592 |