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

Abstract We provide a direct connection between the $\mathcal{Z}_{\max}$ (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 $\mathcal{Z}_{\max}$ -JSJ decomposition of such groups.