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

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.
Comment: 26 pages, 18 figures, revised version