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${l}^{2}$-invisibility and a class of local similarity groups

Part of: Locally compact groups and their algebras Connections with homological algebra and category theory

Published online by Cambridge University Press:  19 August 2014

Roman Sauer  and
Werner Thumann
Roman Sauer
Affiliation:
Karlsruhe Institute of Technology, Karlsruhe, Germany email roman.sauer@kit.edu
Werner Thumann
Affiliation:
Karlsruhe Institute of Technology, Karlsruhe, Germany email werner.thumann@kit.edu
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Abstract

In this note we show that the members of a certain class of local similarity groups 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 ${F}_{\infty }$, 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 ${F}_{\infty }$.

Keywords

l2-cohomologyThompson’s groupzero-in-the-spectrum conjecture

MSC classification

Secondary: 20J05: Homological methods in group theory 22D10: Unitary representations of locally compact groups
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 10 , October 2014 , pp. 1742 - 1754
Copyright
© The Author(s) 2014 

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