L2-invisibility of symmetric operad groups

TL;DR

This paper proves that under specific conditions, the (co)homology of symmetric operad groups vanishes in all dimensions if it vanishes in dimension zero, with applications to groups like the Brin-Thompson groups.

Contribution

It establishes a homological vanishing result for symmetric operad groups, extending understanding of their (co)homological properties and applications to Thompson-like groups.

Findings

Group (co)homology vanishes in all dimensions under certain conditions

Applicable to $l^2$-homology and group ring coefficients

Explicit vanishing results for Thompson-like groups such as $nV$

Abstract

We show a homological result for the class of planar or symmetric operad groups: We show that 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 $nV$.