Florin Rădulescu
Università di Tor Vergata, Rome, Italy & Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest, Romania

Let $G$ be a discrete group and $H$ a subgroup (of infinite index). We are interested in regularity properties of the quasiregular representation $G/H$ of $G$ into the Calkin algebra of $\ell^2(G/H)$. More generally we are interested in the case when $G/H$ is replaced by a countable set $X$ on which $G$ acts. A well known example when this happens is the case when $G$ is $\Gamma \times \Gamma$ and $H$ is the diagonal subgroup (second variable acting from the right). In this case the biexactness phenomena (and hence Akemann Ostrand property) holds true, as proved by N. Ozawa, G. Skandalis and others, for a large class of groups. In particular, in this case the quasiregular representation, modulo compacts, is weakly contained in the left regular representation of the larger group.

In joint work with Jacopo Bassi we prove that temperedness of the quasiregular representation holds true for $G = SL(3,Z)$ and $H = SL(2,Z)$. We also prove weakly mixing type properties for the left and right representation of $SL(3,Z)$ on $\ell^2(Sl(3,Z))$, modulo the compacts. Also in joint work with Jacopo Bassi we find some situations when Akemann Ostrand phenomena holds for "large" subgroups of $SL(3, Z)\times SL(3, Z)$ acting on $\ell^2(Sl(3, Z)$. An ingredient of the proof is provided by the Furstenberg's theory of quasi projective transformations. The method also applies for $PSL(2, Z[1/p])$ instead of $SL(3, Z)$, ($p$ a prime number).