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Röver's Simple Group Is of Type $F_\infty$

James Belk and Francesco Matucci

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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 Belk and Forrest to prove increasing connectivity.

Article information

Publ. Mat., Volume 60, Number 2 (2016), 501-524.

Received: 19 November 2014
Revised: 29 March 2016
First available in Project Euclid: 11 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20J05: Homological methods in group theory 20E08: Groups acting on trees [See also 20F65]

Thompson's groups Grigorchuk's group finiteness properties polysimplicial complex


Belk, James; Matucci, Francesco. Röver's Simple Group Is of Type $F_\infty$. Publ. Mat. 60 (2016), no. 2, 501--524. doi:10.5565/PUBLMAT_60216_07.

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