The Brin-Thompson groups \(sV\) are of the type \(\text F_\infty\).

*(English)*Zbl 1292.20045The Brin-Thompson group \(sV\), with \(s\geq 2\), is a higher-dimensional generalization of the Thompson group \(V\), acting on an \(s\)-dimensional cube. The paper under review shows that these groups are of type \(F_\infty\), i.e. they admit a classifying space with finitely many cells in each dimension. This result was previously known only for \(s\leq 3\) [D. H. Kochloukova et al., Proc. Edinb. Math. Soc., II. Ser. 56, No. 3, 777-804 (2013; Zbl 1294.20065)].

The proof, similarly as in the case of \(s\leq 3\), is based on Brown’s criterion [K. S. Brown, J. Pure Appl. Algebra 44, 45-75 (1987; Zbl 0613.20033)], which states that if a group \(G\) acts on a contractible CW-complex \(X\) with a filtration \(\{X_j\}_{j\geq 1}\) in such a way that \(X_j/G\) are finite, the connectivity of the pair \((X_{j+1},X_j)\) tends to \(\infty\), and the stabilizers of cells are of type \(F_\infty\), then \(G\) itself is of type \(F_\infty\). Kochloukova et al. used the action of \(sV\) on the geometric realization of a certain natural poset, which becomes increasingly difficult to analyze with growing \(s\). The authors of the paper overcome this difficulty by finding a subcomplex \(sX\) of this realization, which they call the Stein space. The advantage they obtain is easier discrete Morse theory, which is used to establish a lower bound on connectivity.

The proof, similarly as in the case of \(s\leq 3\), is based on Brown’s criterion [K. S. Brown, J. Pure Appl. Algebra 44, 45-75 (1987; Zbl 0613.20033)], which states that if a group \(G\) acts on a contractible CW-complex \(X\) with a filtration \(\{X_j\}_{j\geq 1}\) in such a way that \(X_j/G\) are finite, the connectivity of the pair \((X_{j+1},X_j)\) tends to \(\infty\), and the stabilizers of cells are of type \(F_\infty\), then \(G\) itself is of type \(F_\infty\). Kochloukova et al. used the action of \(sV\) on the geometric realization of a certain natural poset, which becomes increasingly difficult to analyze with growing \(s\). The authors of the paper overcome this difficulty by finding a subcomplex \(sX\) of this realization, which they call the Stein space. The advantage they obtain is easier discrete Morse theory, which is used to establish a lower bound on connectivity.

Reviewer: Łukasz Garncarek (Warszawa)

##### MSC:

20F65 | Geometric group theory |

20E32 | Simple groups |

20J05 | Homological methods in group theory |

57Q12 | Wall finiteness obstruction for CW-complexes |

57M07 | Topological methods in group theory |