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

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
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