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

TL;DR

This paper proves that groups discriminated by certain hyperbolic groups are effectively coherent, providing algorithms for subgroup presentation computation, embedding, and discrimination decision, extending to groups with torsion.

Contribution

It establishes effective coherence for groups discriminated by locally quasi-convex hyperbolic groups and develops algorithms for subgroup presentation and discrimination.

Findings

Groups discriminated by locally quasi-convex hyperbolic groups are effectively coherent.

Algorithms for computing subgroup presentations from generators are provided.

Decidability results for discrimination by hyperbolic groups are established.

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.