Detecting geometric splittings in finitely presented groups

TL;DR

This paper introduces algorithms to detect essential free and elementary splittings in finitely presented groups, leveraging solutions to the word problem and acylindricity constants, with applications in geometric group theory.

Contribution

It provides the first algorithms for detecting geometric splittings in groups without 2-torsion, advancing computational methods in group decomposition problems.

Findings

Algorithm determines essential free decompositions.

Algorithm detects elementary splittings in relatively hyperbolic groups.

Applications to computational group theory and geometric decompositions.

Abstract

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant ${\kappa}$, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if $G$ admits an essential free decomposition, the second is an algorithm which; if $G$ is relatively hyperbolic; decides if it admits an essential elementary splitting.