Akemann - Ostrand property for ${\rm PGL}_2(\mathbb Z[\frac{1}{p}])$ relative to ${\rm PSL}_2(\mathbb Z)$

We generalize the Akemann - Ostrand theorem for ${\rm PSL}_2(\mathbb Z)$, to the case of the partial transformations action of ${\rm PGL}_2(\mathbb Z[\frac{1}{p}]) \times {\rm PGL}_2(\mathbb Z[\frac{1}{p}])^{\rm op},$ by left and and right multiplication on $PSL_2(\mathbb Z)$.