Ramanujan complexes and Golden Gates in PU(3)

TL;DR

This paper constructs Ramanujan complexes and Golden Gates for PU(3) by developing new arithmetic lattices that satisfy Ramanujan properties in higher dimensions, overcoming previous obstructions from endoscopic lifts.

Contribution

It introduces a novel family of arithmetic lattices acting simply-transitively on Bruhat-Tits buildings for PU(3), ensuring Ramanujan properties without violating conjectures.

Findings

Constructed Ramanujan complexes from PSL_3(F_p) and PSU_3(F_p)

Developed Golden Gates for PU(3)

Overcame obstructions from endoscopic lifts in higher dimensions

Abstract

In a seminal series of papers from the 80's, Lubotzky, Phillips and Sarnak applied the Ramanujan-Petersson Conjecture for $GL_{2}$ (Deligne's theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat-Tits trees associated with $SL_{2}(\mathbb{Q}_{p})$. As a result, they obtained explicit Ramanujan Cayley graphs from $PSL_{2}\left(\mathbb{F}_{p}\right)$, as well as optimal topological generators ("Golden Gates") for the compact Lie group $PU(2)$. In higher dimension, the naive generalization of the Ramanujan Conjecture fails, due to the phenomenon of endoscopic lifts. In this paper we overcome this problem for $PU_{3}$ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat-Tits buildings associated with $SL_{3}(\mathbb{Q}_{p})$ and $SU_{3}(\mathbb{Q}_{p})$, while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from $PSL_{3}(\mathbb{F}_{p})$ and $PSU_{3}(\mathbb{F}_{p})$, as well as golden gates for $PU(3)$.