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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
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##### References:
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