×

zbMATH — the first resource for mathematics

Cohomological finiteness properties of the Brin-Thompson-Higman groups \(2V\) and \(3V\). (English) Zbl 1294.20065
The 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 \(|K_Y|\).

MSC:
20J05 Homological methods in group theory
57M07 Topological methods in group theory
20F65 Geometric group theory
20E32 Simple groups
PDF BibTeX XML Cite
Full Text: DOI Link arXiv
References:
[1] DOI: 10.1007/s10711-009-9423-9 · Zbl 1213.20029 · doi:10.1007/s10711-009-9423-9
[2] Queen Mary College Mathematical Notes (1981)
[3] DOI: 10.1007/s002220050168 · Zbl 0888.20021 · doi:10.1007/s002220050168
[4] Cambridge Studies in Advanced Mathematics 31 (1998)
[5] Notes on Pure Mathematics 8 (1974)
[6] DOI: 10.1007/s10711-004-8122-9 · Zbl 1136.20025 · doi:10.1007/s10711-004-8122-9
[7] DOI: 10.2140/pjm.2012.257.53 · Zbl 1248.20035 · doi:10.2140/pjm.2012.257.53
[8] Graduate Texts in Mathematics 78 (1981)
[9] DOI: 10.1016/0022-4049(87)90015-6 · Zbl 0613.20033 · doi:10.1016/0022-4049(87)90015-6
[10] DOI: 10.1016/j.jalgebra.2004.10.028 · Zbl 1135.20022 · doi:10.1016/j.jalgebra.2004.10.028
[11] Mathematics and Its Applications 6 (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.