Ramanujan Complexes and Golden Gates in PU(3)

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).