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

Abstract

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 ${\mathrm{PGL}}_2(\mathbb{Z}[\frac{1}{p}])\times {\mathrm{PGL}}_2(\mathbb{Z}[\frac{1}{p}]{)}^{\mathrm{op}},$ on ${\ell }^2({\mathrm{PSL}}_2(\mathbb{Z}))$, 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

arXiv commentThis article contains the technical part (the second part) of the article arXiv:0802.3548. It has been rewritten as a separate paper. Some parts of proofs have been revised