Finiteness properties of simple groups

A group is said to be of type $F_n$ if it admits a classifying space with compact $n$-skeleton. We will consider the class of R\"{o}ver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type $F_{n-1}$ but not $F_n$ for each $n$. These are the first known examples for $n\geq 3$.
As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups, the first is due to Caprace and R\'{e}my.  This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.