#### Volume 16, issue 4 (2016)

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$L^2$–invisibility of symmetric operad groups

### Werner Thumann

Algebraic & Geometric Topology 16 (2016) 2229–2255
##### Abstract

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 $nV$.

##### Keywords
operad groups, Thompson groups, group homology, L2-homology
##### Mathematical Subject Classification 2010
Primary: 20J05
Secondary: 22D10, 18D50