On the representation of a discrete group $Γ$ with subgroup $Γ_0$ in the Calkin algebra of $\ell^2(Γ/ Γ_0)$

Details On the representation of a discrete group $Γ$ with subgroup $Γ_0$ in the Calkin algebra of $\ell^2(Γ/ Γ_0)$ On the representation of a discrete group $Γ$ with subgroup $Γ_0$ in the Calkin algebra of $\ell^2(Γ/ Γ_0)$

2011-05-24

arxiv.org/abs/1105.4854v4

Mathematics

Group Theory

The argument for the case of no fixed points in the proof of main theorem has been changed along the lines of the additional a

Scientific paper

Given a countable group $\tilde{\Gamma}$ with an infinite, proper subgroup $\Gamma_0$, we analyze the ergodical properties of the action of $\tilde{\Gamma}$ on the infinite sets consisting of cosets of stabilizers for the action of $\tilde{\Gamma}$ on $\Gamma/\Gamma_0$, modulo finite sets. In this way we find sufficient conditions such that the representation of $\tilde{\Gamma}$ on the Calkin algebra of $\ell^2(\tilde{\Gamma}/\Gamma_0)$ be contained in the left regular representation of $\tilde{\Gamma}$. In the case $\Gamma$ is a discrete group, and $\tilde{\Gamma}=\Gamma\times\Gamma^{\rm op}$ and $\Gamma_0=\{(\gamma,\gamma^{-1}\mid\gamma\in\Gamma\}$ we recover the Akemann-Ostrand property for $\Gamma\times\Gamma^{\rm op}$ acting by left and right convolution on $\Gamma$. We verify the sufficient conditions for $\Gamma=\PSL_2(\Z[\frac1p])$, and for $\Gamma=\SL_3(\Z)$. Consequently, these groups have the AO property. This implies, using the solidity property of Ozawa ([Oz]), that the for the corresponding group von Neumann algebras, we have: $\L(\SL_3(\Z))\not\cong \L(\SL_n(\Z))$, $n\geq 4$.

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