# Algorithms detecting stability and Morseness for finitely generated groups

## TL;DR

This paper develops algorithms to detect stability and Morseness of finitely generated subgroups across various group classes, extending previous work on hyperbolic groups to broader contexts.

## Contribution

It introduces new detection and decidability algorithms for stability and Morseness in diverse finitely generated groups beyond hyperbolic groups.

## Key findings

- Algorithms successfully detect stability and Morseness in mapping class groups
- Decidability results extend to right-angled Artin groups and toral relatively hyperbolic groups
- Applicable to groups discriminated by hyperbolic groups, including limit groups

## Abstract

The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which, on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup of $G$ and runs forever otherwise. In this paper, we give various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups, and finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups).