Open Access
October, 2007 On vanishing of $L^{2}$-Betti numbers for groups
Jang Hyun JO
J. Math. Soc. Japan 59(4): 1031-1044 (October, 2007). DOI: 10.2969/jmsj/05941031

Abstract

We show that if a group $G$ admits a finite dimensional contractible $G$-CW-complex $X$ then the vanishing of the $L^2$-Betti numbers for all stabilizers $G_\sigma$ of $X$ determines that of the $L^2$-Betti numbers for $G$. We also give a relation among the $L^2$-Euler characteristics for $X$ as a $G$-CW-complex and those for $X$ as a $G_\sigma$-CW-complex under certain assumptions. Finally, we present a new class of groups satisfying the Chatterji-Mislin conjecture which amounts to putting Brown's formula within the framework of $L^2$-homology.

Citation

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Jang Hyun JO. "On vanishing of $L^{2}$-Betti numbers for groups." J. Math. Soc. Japan 59 (4) 1031 - 1044, October, 2007. https://doi.org/10.2969/jmsj/05941031

Information

Published: October, 2007
First available in Project Euclid: 10 December 2007

zbMATH: 1136.55005
MathSciNet: MR2370004
Digital Object Identifier: 10.2969/jmsj/05941031

Subjects:
Primary: 55N99
Secondary: 20F65 , 46L99

Keywords: $\mathcal{C}$-exact , $L^{2}$-Betti number , $L^{2}$-Euler characteristic , Amenable , weak $\mathcal{C}$-exact

Rights: Copyright © 2007 Mathematical Society of Japan

Vol.59 • No. 4 • October, 2007
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