Algorithmic Properties of Relatively Hyperbolic Groups

By Donovan Rebbechi. 2003

53 citations recorded by the source corpus.

The following discourse is inspired by the works on hyperbolic groups of Epstein, and Neumann/Reeves. Epstein showed that geometrically finite hyperbolic groups are biautomatic. Neumann/Reeves showed that virtually central extensions of word hyperbolic groups are biautomatic. We prove the following generalisation: Theorem. Let H be a geometrically finite hyperbolic group. Let sigma in H^2(H) and suppose that sigma restricted to P is zero for any parabolic subgroup P of H. Then the extension of H by sigma is biautomatic. We also prove another generalisation of the result of Epstein. Theorem. Let G be hyperbolic relative to H, with the bounded coset penetration property. Let H be a biautomatic group with a prefix-closed normal form. Then G is biautomatic. Based on these two results, it seems reasonable to conjecture the following (which the author believes can be proven with a simple generalisation of the argument in Section 1): Let G be hyperbolic relative to H, where H has a prefixed closed biautomatic structure. Let sigma in H^2(G) and suppose that sigma restricted to H is zero. Then the extension of G by sigma is biautomatic.

Subfields: Computational Theory and Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1dce2d9a84ad0c8def273a415e583168d6e8c656

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In Computational Theory and Mathematics

1. Franccois Dahmani
2. Vincent Guirardel
3. J. Manning
4. Christopher Hruska
5. Bruce Kleiner
6. Denis V. Osin
7. D. Groves
8. Emina Alibegović

In Computer Graphics and Computer-Aided Design

1. J. Manning
2. Christopher Hruska
3. Bruce Kleiner
4. Denis V. Osin
5. D. Groves
6. Emina Alibegović

In Geometry and Topology

1. D. Groves
2. Denis V. Osin
3. Vincent Guirardel
4. Alessandro Sisto
5. J. Manning
6. Christopher Hruska
7. Franccois Dahmani
8. Emina Alibegović

Citations in the corpus

• Dehn filling in relatively hyperbolic groups (2006, background): “There are a number of previous constructions of cusped spaces (for example: Cannon and Cooper [7], Bowditch [4] and Rebbechi [30].”
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (2011, background): “e most suitable one for our purposes. That relative hyperbolicity in the sense of [32, 61, 67] implies relative hyperbolicity in the sense of the denition stated below is essentially due to Rebbechi [133]. Indeed it was proved in [133] for nitely presented groups. The later condition is not really i”
• Hadamard spaces with isolated flats (2004, methodology): “We make use of these formulations in the proof of Theorem 5.2.5, which establishes the last remaining equivalence (1) ⇐⇒ (2) of Theorem 1.2.1.”
• A Combination Theorem for Relatively Hyperbolic Groups (2003, background): “D. Y. Rebecchi showed that a group G hyperbolic relative to a biautomatic subgroup H is itself biautomatic, [ 7 ].”
• Relative hyperbolicity and relative quasiconvexity for countable groups (2008, unknown): “The idea of gluing horoballs onto a nitely generated group is due to Cannon{Cooper [CC92], and variations have been studied by Bowditch, Rebbechi, and Groves{Manning [Bow99, Reb01 , GM].”