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 $G{L}_2$ (Deligne's theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat-Tits trees associated with $S{L}_2({\mathbb{Q}}_p)$. As a result, they obtained explicit Ramanujan Cayley graphs from $PS{L}_2\left({\mathbb{F}}_p\right)$, 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 $P{U}_3$ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat-Tits buildings associated with $S{L}_3({\mathbb{Q}}_p)$ and $S{U}_3({\mathbb{Q}}_p)$, while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from $PS{L}_3({\mathbb{F}}_p)$ and $PS{U}_3({\mathbb{F}}_p)$, as well as golden gates for $PU(3)$.

arXiv commentTo appear in Geometric and Functional Analysis

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Cited by in corpus (2)

• Cutoff on Ramanujan complexes and classical groups
• Free flags over local rings and powering of high dimensional expanders