Hyperbolic one-relator groups

TL;DR

This paper introduces primitive extension groups and demonstrates that hyperbolicity of one-relator groups can be characterized by the hyperbolicity of these subgroups, providing explicit decompositions and new structural insights.

Contribution

It establishes a reduction of hyperbolic one-relator groups to primitive extension groups and characterizes these groups using Christoffel words and graph of free groups decompositions.

Findings

Hyperbolic one-relator groups have quasi-convex Magnus subgroups.

Primitive extension groups admit explicit graph-of-groups decompositions.

Characterization of 2-free one-relator groups with exceptional intersection.

Abstract

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterising hyperbolic one-relator groups to characterising hyperbolic primitive extension groups. These new groups moreover admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterise $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroups and build upon the one-relator tower machinery developed in the authors previous article.