JSJ decompositions and polytopes for two-generator one-relator groups

TL;DR

This paper establishes a direct link between JSJ decompositions and Friedl--Tillmann polytopes for certain hyperbolic two-generator one-relator groups, enabling efficient algorithms for their structural analysis.

Contribution

It introduces a novel connection between JSJ decompositions and polytopes, leading to new structural insights and a quadratic-time algorithm for computing the Z_{max}-JSJ decomposition.

Findings

Established a direct connection between JSJ decompositions and Friedl--Tillmann polytopes.

Proved the existence of a quadratic-time algorithm for Z_{max}-JSJ decomposition.

Provided structural and algorithmic properties of the groups studied.

Abstract

We provide a direct connection between the Z_{max} (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank $2$. We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the Z_{max}-JSJ decomposition of such groups.