Cohomological finiteness properties of the Brin-Thompson-Higman groups 2V and 3V

TL;DR

This paper proves that the groups 2V and 3V, generalizations of Thompson-Higman groups, possess strong finiteness properties and are finitely presented, using novel methods.

Contribution

It establishes that 2V and 3V are of type FP_infinity and provides a new proof of their finite presentability.

Findings

2V and 3V are of type FP_infinity

Both groups are finitely presented

New proof technique for finite presentability

Abstract

We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented.