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${l}^{2}$-invisibility and a class of local similarity groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 10 / October 2014
- Published online by Cambridge University Press:
- 19 August 2014, pp. 1742-1754
- Print publication:
- October 2014
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- Article
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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 
${F}_{\infty }$.
