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

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