The agrarian polytope of two-generator one-relator groups

TL;DR

This paper proves that the agrarian polytope for certain two-generator one-relator groups is invariant under presentation changes and encodes the group's splitting complexity when torsion-free, extending previous theorems.

Contribution

It confirms the invariance of the agrarian polytope for these groups and links it to splitting complexity in torsion-free cases, generalizing earlier results.

Findings

The agrarian polytope is independent of presentation for these groups.

In torsion-free cases, the polytope encodes splitting complexity.

The work generalizes previous theorems by Friedl-Tillmann and Friedl-Lück-Tillmann.

Abstract

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-L\"uck-Tillmann.