\(L^2\)-invisibility of symmetric operad groups. (English) Zbl 1387.20042

Thumann, Werner

Algebr. Geom. Topol. 16, No. 4, 2229-2255 (2016).

Summary: We show a homological result for the class of planar or symmetric operad groups: under certain conditions, group (co)homology of such groups with certain coefficients vanishes in all dimensions, provided it vanishes in dimension 0. This can be applied, for example, to \(l^2\)-homology or cohomology with coefficients in the group ring. As a corollary, we obtain explicit vanishing results for Thompson-like groups such as the Brin-Thompson groups \(nV\).

MSC:

20J05	Homological methods in group theory
22D10	Unitary representations of locally compact groups
18D50	Operads (MSC2010)

Keywords:

operad groups; Thompson groups; group homology; \(L^2\)-homology