${l}^{2}$-invisibility and a class of local similarity groups. (19th August 2014)

• Record Type: Journal Article
• Authors: Sauer, Roman; Thumann, Werner
• Is Part Of: Compositio mathematica. Volume 150:Number 10(2014)
• Page Start: 1742
• Page End: 1754
• Publication Date: 2014-08-19
• Subjects: 20J05 (primary), -- 22D10 (secondary); l2-cohomology, -- Thompson's group, -- zero-in-the-spectrum conjecture; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X14007313
• Languages: English
• ISSNs: 0010-437X

Abstract

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 ${F}_{\infty }$ .