Hyperbolic hydra. (English) Zbl 1318.20041
A previous contribution of T. R. Riley and W. Dison on the same topic can be found in [Comment. Math. Helv. 88, No. 3, 507-540 (2013; Zbl 1305.20052)].
The present paper deals with another interesting aspect of Hydra groups. Specifically, the authors prove that:
Theorem 1.1. There are hyperbolic groups \(\Gamma_k\) for all \(k\geq 1\) with free rank \((k+18)\) subgroups \(\Lambda_k\) whose distortion satisfies \(\mathrm{Dist}^{\Gamma_k}_{\Lambda_k}\geq A_k\), that is, grows at least like the \(k\)-th Ackermann function \(A_k\).
This is the first time that we may note the above lower bound for the subgroup distortion of hyperbolic groups of large free rank.
The present paper deals with another interesting aspect of Hydra groups. Specifically, the authors prove that:
Theorem 1.1. There are hyperbolic groups \(\Gamma_k\) for all \(k\geq 1\) with free rank \((k+18)\) subgroups \(\Lambda_k\) whose distortion satisfies \(\mathrm{Dist}^{\Gamma_k}_{\Lambda_k}\geq A_k\), that is, grows at least like the \(k\)-th Ackermann function \(A_k\).
This is the first time that we may note the above lower bound for the subgroup distortion of hyperbolic groups of large free rank.
Reviewer: Francesco G. Russo (Rondebosch)
MSC:
20F65 | Geometric group theory |
20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
20F67 | Hyperbolic groups and nonpositively curved groups |
20F05 | Generators, relations, and presentations of groups |
20E07 | Subgroup theorems; subgroup growth |
Keywords:
hyperbolic groups; subgroup distortion; Hydra groups; Ackermann functions; free subgroups of finite rank; presentationsCitations:
Zbl 1305.20052References:
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