Non-orientable surface-plus-one-relation groups

TL;DR

This paper extends algebraic methods to analyze the $L^2$-Betti numbers of non-orientable surface-plus-one-relation groups, building on prior topological results for orientable cases.

Contribution

It generalizes existing topological results to a broader class of two-relator groups using algebraic techniques and computes their $L^2$-Betti numbers.

Findings

Extended results of Hempel and Howie to non-orientable groups

Determined $L^2$-Betti numbers for non-orientable surface-plus-one-relation groups

Unified algebraic approach for two-relator groups

Abstract

Recently Dicks-Linnell determined the $L^2$-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the $L^2$-Betti numbers of the non-orientable surface-plus-one-relation groups.