TY - JOUR
T1 - Ramanujan Complexes and Golden Gates in PU(3)
AU - Evra, Shai
AU - Parzanchevski, Ori
N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2022/4
Y1 - 2022/4
N2 - In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for GL2 (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with SL2(Qp). As a result, they obtained explicit Ramanujan Cayley graphs from PSL2(Fp) , 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 PU3 by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with SL3(Qp) and SU3(Qp) , while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from PSL3(Fp) and PSU3(Fp) , as well as golden gates for PU(3).
AB - In a seminal series of papers from the 80’s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for GL2 (Deligne’s theorem), to a special family of arithmetic lattices, which act simply-transitively on the Bruhat–Tits trees associated with SL2(Qp). As a result, they obtained explicit Ramanujan Cayley graphs from PSL2(Fp) , 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 PU3 by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with SL3(Qp) and SU3(Qp) , while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from PSL3(Fp) and PSU3(Fp) , as well as golden gates for PU(3).
UR - http://www.scopus.com/inward/record.url?scp=85127280394&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s00039-022-00593-9
DO - https://doi.org/10.1007/s00039-022-00593-9
M3 - مقالة
SN - 1016-443X
VL - 32
SP - 193
EP - 235
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 2
ER -