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:
| [1] | G. N. Arzhantseva and D. V. Osin, Solvable groups with polynomial Dehn functions. Trans. Amer. Math. Soc. 354 (2002), 3329-3348. · Zbl 0998.20040 · doi:10.1090/S0002-9947-02-02985-9 |
| [2] | J. Barnard and N. Brady, Distortion of surface groups in CAT.0/ free-by-cyclic groups. Geom. Dedicata 120 (2006), 119-139. · Zbl 1167.20024 · doi:10.1007/s10711-006-9072-1 |
| [3] | J. Barnard, N. Brady, and P. Dani, Super-exponential distortion of subgroups of CAT. 1/ groups. Algebr. Geom. Topol. 7 (2007), 301-308. · Zbl 1187.20055 · doi:10.2140/agt.2007.7.301 |
| [4] | M. R. Bridson, On the existence of flat planes in spaces of non-positive curvature. Proc. Amer. Math. Soc. 123 (1995), 223-235. · Zbl 0840.53033 · doi:10.2307/2160630 |
| [5] | M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature . Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999. · Zbl 0988.53001 |
| [6] | W. Dison and T. Riley, Hydra groups. Comment. Math. Helv. 88 (2013), 507-540. · Zbl 1305.20052 · doi:10.4171/CMH/294 |
| [7] | B. Farb, The extrinsic geometry of subgroups and the generalized word problem. Proc. London Math. Soc. (3) 68 (1994), 577-593. · Zbl 0816.20032 · doi:10.1112/plms/s3-68.3.577 |
| [8] | M. Gromov, Hyperbolic groups. In Essays in group theory , Math. Sci. Res. Inst. Publ. 8, Springer-Verlag, New York 1987, 75-263. · Zbl 0634.20015 |
| [9] | M. Gromov, Geometric group theory (Sussex, 1991), vol. 2: Asymptotic invariants of infinite groups. London Math. Soc. Lecture Note Ser. 182, Cambridge University Press, Cambridge 1993. · Zbl 0841.20039 |
| [10] | J. Howie, On the asphericity of ribbon disc complements. Trans. Amer. Math. Soc. 289 (1985), 281-302. · Zbl 0572.57001 · doi:10.2307/1999700 |
| [11] | T. Mecham and A. Mukherjee, Hyperbolic groups which fiber in infinitely many ways. Algebr. Geom. Topol. 9 (2009), 2101-2120. · Zbl 1223.20036 · doi:10.2140/agt.2009.9.2101 |
| [12] | M. Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces. J. Differential Geom. 48 (1998), 135-164. · Zbl 0906.20023 |
| [13] | M. Mitra, Coarse extrinsic geometry: a survey. In The Epstein birthday schrift , Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry 1998, 341-364. · Zbl 0914.20034 |
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