On Residualizing Homomorphisms Preserving Quasiconvexity

TL;DR

This paper investigates the properties of certain subgroups in hyperbolic groups, showing that under specific conditions, the finiteness requirement can be replaced by quasiconvexity, extending Ol'shanskii's earlier work.

Contribution

It generalizes Ol'shanskii's description of G-subgroups by replacing the finiteness condition with quasiconvexity under natural assumptions.

Findings

Finiteness assumption can be replaced by quasiconvexity for G-subgroups.

Extends Ol'shanskii's classification to broader conditions.

Provides new insights into subgroup structures in hyperbolic groups.

Abstract

$H$ is called a $G$-subgroup of a hyperbolic group $G$ if for any finite subset $M\subset G$ there exists a homomorphism from $G$ onto a non-elementary hyperbolic group $G_1$ that is surjective on $H$ and injective on $M$. In his paper in 1993 A. Ol'shanskii gave a description of all $G$-subgroups in any given non-elementary hyperbolic group $G$. Here we show that for the same class of $G$-subgroups the finiteness assumption on $M$ (under certain natural conditions) can be replaced by an assumption of quasiconvexity.