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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.

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