Röver’s simple group is of type \(F_\infty\). (English) Zbl 1376.20043

Summary: 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 polysimplicial complex analogous to the Farley complex for \(V\). We then analyze the descending links of the polysimplicial complex, using a theorem of J. Belk and B. Forrest [“Rearrangement groups of fractals”, Preprint, arXiv:1510.03133] to prove increasing connectivity.


20F65 Geometric group theory
20E32 Simple groups
20J05 Homological methods in group theory
20E08 Groups acting on trees
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