Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

TL;DR

This paper introduces a novel method inspired by Stallings' foldings to construct non-quasiconvex subgroups within hyperbolic right-angled Coxeter groups, providing explicit examples with various properties.

Contribution

It presents a new technique for constructing non-quasiconvex subgroups of hyperbolic groups, especially within RACGs, expanding the toolkit for subgroup analysis in geometric group theory.

Findings

Constructed explicit non-quasiconvex subgroups with short generators

Generated finitely generated free non-quasiconvex subgroups

Produced non-finitely presentable subgroups in hyperbolic RACGs

Abstract

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2-dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.