Cohomological finiteness properties of the Brin-Thompson-Higman groups \(2V\) and \(3V\).

*(English)*Zbl 1294.20065The main result of the article is that Brin’s higher-dimensional versions \(2V\) and \(3V\) of Thompson’s group \(V\) are of type \(F_\infty\).

The general strategy follows K. S. Brown [J. Pure Appl. Algebra 44, 45-75 (1987; Zbl 0613.20033)] and is by now fairly standard: each of the groups naturally acts with finite stabilizers on a poset \(\mathfrak A\) of “admissible subsets”. The realization \(|\mathfrak A|\) of this poset is contractible and admits a cocompact filtration \(|\mathfrak A_n|\). Using Brown’s criterion, this reduces the problem to showing that the descending links \(K_Y\) are asymptotically highly connected.

As usual, this is the crux of the problem, and it is solved (for \(2V\)) in Section 4 of the article. The main steps are to first show that for every \(t\) there is a subcomplex \(\Sigma_t\) such that the \(t\)-skeleton of \(|K_Y|\) can be homotoped into \(\Sigma_{4t}\) (this is done in Section 4.2); and secondly to show that \(\Sigma_t\) can be collapsed within \(|K_Y|\) provided \(Y\) is large enough (Section 4.3).

For \(3V\) the only thing that needs a new proof is that the \(t\)-skeleton can be homotoped into \(\Sigma_{8t}\), which is shown in Section 5. It becomes apparent that pursuing this approach in higher dimension would become increasingly involved.

The result was generalized to all Brin-Thompson groups \(sV\) by M. G. Fluch et al., [Pac. J. Math. 266, No. 2, 283-295 (2013; Zbl 1292.20045)]. The main difference there is the use of the Stein-Farley complex, which is a subspace of \(|K_Y|\), instead of all of \(|K_Y|\).

The general strategy follows K. S. Brown [J. Pure Appl. Algebra 44, 45-75 (1987; Zbl 0613.20033)] and is by now fairly standard: each of the groups naturally acts with finite stabilizers on a poset \(\mathfrak A\) of “admissible subsets”. The realization \(|\mathfrak A|\) of this poset is contractible and admits a cocompact filtration \(|\mathfrak A_n|\). Using Brown’s criterion, this reduces the problem to showing that the descending links \(K_Y\) are asymptotically highly connected.

As usual, this is the crux of the problem, and it is solved (for \(2V\)) in Section 4 of the article. The main steps are to first show that for every \(t\) there is a subcomplex \(\Sigma_t\) such that the \(t\)-skeleton of \(|K_Y|\) can be homotoped into \(\Sigma_{4t}\) (this is done in Section 4.2); and secondly to show that \(\Sigma_t\) can be collapsed within \(|K_Y|\) provided \(Y\) is large enough (Section 4.3).

For \(3V\) the only thing that needs a new proof is that the \(t\)-skeleton can be homotoped into \(\Sigma_{8t}\), which is shown in Section 5. It becomes apparent that pursuing this approach in higher dimension would become increasingly involved.

The result was generalized to all Brin-Thompson groups \(sV\) by M. G. Fluch et al., [Pac. J. Math. 266, No. 2, 283-295 (2013; Zbl 1292.20045)]. The main difference there is the use of the Stein-Farley complex, which is a subspace of \(|K_Y|\), instead of all of \(|K_Y|\).

Reviewer: Stefan Witzel (Bielefeld)

##### MSC:

20J05 | Homological methods in group theory |

57M07 | Topological methods in group theory |

20F65 | Geometric group theory |

20E32 | Simple groups |

##### Keywords:

Brin-Thompson-Higman groups; topological finiteness properties of groups; generalized Thompson groups; cohomological finiteness properties##### References:

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