L2-invisibility and a class of local similarity groups

Abstract

In this note we show that the members of a certain class of local similarity groups are l2-invisible, i.e. the non-reduced group homology of the regular unitary representation vanishes in all degrees. This class contains for example Thompson's group V and Nekrashevych-R\"over groups. They yield counterexamples to a generalized zero-in-the-spectrum conjecture for groups that admit a classifying space with finitely many cells in each dimension.