Röver's Simple Group is of Type $F_{\infty }$

Abstract

We prove that Claas Röver's Thompson-Grigorchuk simple group $V\mathcal{G}$ has type $F_{\infty }$. The proof involves constructing two complexes on which $V\mathcal{G}$ acts: a simplicial complex analogous to the Stein complex for $V$, and a polysimiplical complex analogous to the Farley complex for $V$. We then analyze the descending links of the polysimplicial complex, using a theorem of Belk and Forrest to prove increasing connectivity.

18 pages, no figures; improved exposition and added details

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