L2-invisibility and a class of local similarity groups

In this talk we present a certain class of local similarity groups which are L^2-invisible, i.e. the (non-reduced) group homology of the regular unitary representation vanishes in all degrees. This class contains groups of type FP_\infty, e.g. Thompson's group V and Nekrashevych-Röver groups. They yield counterexamples to a generalized zero-in-the-spectrum conjecture for groups of type FP_\infty. This is joint work with Roman Sauer.