Random quotients preserve acylindrical and hierarchical hyperbolicity
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910001458577408 |
|---|---|
| author | Abbott, Carolyn Berlyne, Daniel Mangioni, Giorgio Ng, Thomas Rasmussen, Alexander J. |
| author_facet | Abbott, Carolyn Berlyne, Daniel Mangioni, Giorgio Ng, Thomas Rasmussen, Alexander J. |
| contents | We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random quotients preserve acylindrical and hierarchical hyperbolicity Abbott, Carolyn Berlyne, Daniel Mangioni, Giorgio Ng, Thomas Rasmussen, Alexander J. Group Theory Geometric Topology 20F65 (Primary) 20F67, 05C81 (Secondary) We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main tools relate the theories of spinning families and projection complexes to random walks. In the presence of a hierarchical hyperbolic structure on the group, we leverage the fine control of projections to show that this structure is preserved in the quotient asymptotically almost surely. The same techniques yield that random quotients of a non-elementary hyperbolic group (relative to any finite collection of finitely generated peripheral subgroups) are asymptotically almost surely hyperbolic (relative to commensurable peripheral subgroups). Finally, we also prove that any two groups that are both acylindrically and hierarchically hyperbolic have a common quotients which is itself acylindrically and hierarchically hyperbolic. This produces "exotic" hierarchically hyperbolic groups with strong fixed point properties, such as Kazhdan's property (T). |
| title | Random quotients preserve acylindrical and hierarchical hyperbolicity |
| topic | Group Theory Geometric Topology 20F65 (Primary) 20F67, 05C81 (Secondary) |
| url | https://arxiv.org/abs/2507.16677 |