Journal of the Mathematical Society of Japan

On vanishing of $L^{2}$-Betti numbers for groups

Jang Hyun JO

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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.

Article information

J. Math. Soc. Japan, Volume 59, Number 4 (2007), 1031-1044.

First available in Project Euclid: 10 December 2007

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Zentralblatt MATH identifier

Primary: 55N99: None of the above, but in this section
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 46L99: None of the above, but in this section

amenable $L^{2}$-Betti number $L^{2}$-Euler characteristic $\mathcal{C}$-exact weak $\mathcal{C}$-exact


JO, Jang Hyun. On vanishing of $L^{2}$-Betti numbers for groups. J. Math. Soc. Japan 59 (2007), no. 4, 1031--1044. doi:10.2969/jmsj/05941031.

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