Groups, Geometry, and Dynamics


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Volume 10, Issue 2, 2016, pp. 545–582
DOI: 10.4171/GGD/356

Published online: 2016-06-09

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

Inna Bumagin[1] and Jeremy Macdonald[2]

(1) Carleton University, Ottawa, Canada
(2) Stevens Institute of Technology, Hoboken, USA

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.

Keywords: Hyperbolic groups, quasi-convexity, discrimination, subgroup presentations, algorithms

Bumagin Inna, Macdonald Jeremy: Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group. Groups Geom. Dyn. 10 (2016), 545-582. doi: 10.4171/GGD/356