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

TL;DR

This paper extends the Akemann-Ostrand theorem to certain group actions involving PGL groups over localized integers, and applies this to analyze the spectrum of Hecke operators related to Ramanujan conjectures.

Contribution

It generalizes the Akemann-Ostrand property to PGL groups with cocycles and applies this to spectral bounds of Hecke operators in number theory.

Findings

Extended Akemann-Ostrand theorem to PGL groups with cocycles.

Proved the essential spectrum of Hecke operators lies within specific bounds.

Connected operator algebra techniques with Ramanujan conjecture analysis.

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 ${\rm PGL}_2(\mathbb Z[\frac{1}{p}]) \times {\rm PGL}_2(\mathbb Z[\frac{1}{p}])^{\rm op},$ 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