R\"over's Simple Group is of Type $F_\infty$

TL;DR

This paper proves that R"over's simple group $V\mathcal{G}$ has type $F_\infty$, demonstrating its high finiteness properties through complex constructions and connectivity analysis.

Contribution

It introduces new complexes for $V\mathcal{G}$ and proves their high connectivity, establishing the group's type $F_\infty$ status.

Findings

$V\mathcal{G}$ has type $F_\infty$

Construction of complexes analogous to known complexes for $V$

Analysis of descending links using Belk and Forrest's theorem

Abstract

We prove that Claas R\"over'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.