Ramanujan Complexes and Golden Gates in PU (3)

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(ℚ_p) . As a result, they obtained explicit Ramanujan Cayley graphs from PSL_2( 𝔽_p) , 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(ℚ_p) and SU_3(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(𝔽_p) and PSU_3(𝔽_p) , as well as golden gates for PU (3).