L2-invisibility and a class of local similarity groups

TL;DR

This paper demonstrates that certain local similarity groups, including Thompson's group V and Nekrashevych-Röver groups, are L2-invisible, providing counterexamples to a generalized spectral conjecture.

Contribution

It introduces a class of local similarity groups that are L2-invisible and shows they serve as counterexamples to a spectral conjecture.

Findings

Members of the class are L2-invisible in all degrees.

Includes Thompson's group V and Nekrashevych-Röver groups.

Counterexamples to a generalized zero-in-the-spectrum conjecture.

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.