Division in group rings of surface groups

TL;DR

This paper develops a division algorithm for group rings of high genus surface groups and applies it to classify certain 2-complexes and analyze cohomological dimensions of specific groups.

Contribution

It introduces a division algorithm for surface group rings and demonstrates its applications in topology and group cohomology.

Findings

Division algorithm for high genus surface groups

Classification of certain 2-complexes with surface fundamental groups

Bounds on cohomological dimensions of 2-relator groups

Abstract

We prove a division algorithm for group rings of high genus surface groups and use it to show that some $2$-complexes with surface fundamental groups are standard. We also give an application of division to cohomological dimension of $2$-relator groups acting on $\mathbb H^n$.