Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group

DOI10.4171/GGD/356zbMATH Open1383.20032arXiv1307.6783OpenAlexW2962859007MaRDI QIDQ2630763

Jeremy Macdonald, Inna Bumagin

Publication date: 22 July 2016

Published in: Groups, Geometry, and Dynamics (Search for Journal in Brave)

Abstract: We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $Gamma$ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the subgroup generators. We study $G$ via its embedding into an iterated centralizer extension of $Gamma$, and prove that this embedding can be computed. We also give algorithms to enumerate all finitely generated groups discriminated by $Gamma$ and to decide whether a given group, with decidable word problem, is discriminated by $Gamma$. If $Gamma$ may have torsion, we prove that groups obtained from $Gamma$ by iterated amalgamated products with virtually abelian groups, over elementary subgroups, are effectively coherent.

Full work available at URL: https://arxiv.org/abs/1307.6783

zbMATH Keywords

algorithmshyperbolic groupsquasi-convexitydiscriminationsubgroup presentations

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Generators, relations, and presentations of groups (20F05) Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) (20F10) Hyperbolic groups and nonpositively curved groups (20F67) Residual properties and generalizations; residually finite groups (20E26) Local properties of groups (20E25)

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