Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.14387 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917423746121728 |
|---|---|
| author | Legaspi, Xabier Steenbock, Markus |
| author_facet | Legaspi, Xabier Steenbock, Markus |
| contents | An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. There are two consequences: firstly, there is a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. Secondly, the uniform uniform exponential growth rate of a classical $C''(λ)$-small cancellation group, for sufficiently small $λ$, is bounded from below by a universal positive constant. We give a similar result for uniform entropy-cardinality estimates. This yields an explicit upper bound on the isomorphism class of marked $δ$-hyperbolic $C''(λ)$-small cancellation groups of uniformly bounded entropy in terms of $δ$ and the entropy bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniform growth in small cancellation groups Legaspi, Xabier Steenbock, Markus Group Theory An open question asks whether every group acting acylindrically on a hyperbolic space has uniform exponential growth. We prove that the class of groups of uniform uniform exponential growth acting acylindrically on a hyperbolic space is closed under taking certain geometric small cancellation quotients. There are two consequences: firstly, there is a finitely generated acylindrically hyperbolic group that has uniform exponential growth but has arbitrarily large torsion balls. Secondly, the uniform uniform exponential growth rate of a classical $C''(λ)$-small cancellation group, for sufficiently small $λ$, is bounded from below by a universal positive constant. We give a similar result for uniform entropy-cardinality estimates. This yields an explicit upper bound on the isomorphism class of marked $δ$-hyperbolic $C''(λ)$-small cancellation groups of uniformly bounded entropy in terms of $δ$ and the entropy bound. |
| title | Uniform growth in small cancellation groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2405.14387 |