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The Brin-Thompson groups $$sV$$ are of the type $$\text F_\infty$$. (English) Zbl 1292.20045
The 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.

##### 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
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