Twisted Akemann-Ostrand property for ${\mathrm{P}\mathrm{G}\mathrm{L}}_2(\mathbb{Z}[\frac{1}{p}])$ and Ramanujan Petersson Conjectures

We prove an extension of the Akemann - Ostrand theorem, regarding the simultaneous, left and right regular representations of the free group, modulo compact operators, to the case of the partial action of ${\rm PGL}_2(\mathbb Z[\frac{1}{p}]) \times {\rm PGL}_2(\mathbb Z[\frac{1}{p}])^{\rm op},$on$\ell^2({\rm PSL}_2(\mathbb Z))$, in the presence of a non-trivial cocycle. We use this result and the operator algebra techniques developed previously to prove that the essential spectrum of the Hecke operators is contained in the bounds