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 $PU(3)$.