Ramanujan Complexes and Golden Gates in PU(3)

Shai Evra; Ori Parzanchevski

doi:10.1007/s00039-022-00593-9

Ramanujan Complexes and Golden Gates in PU(3)

Shai Evra^*, Ori Parzanchevski

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

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(Q_p). As a result, they obtained explicit Ramanujan Cayley graphs from PSL₂(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 PU₃ by constructing a family of arithmetic lattices which act simply-transitively on the Bruhat–Tits buildings associated with SL₃(Q_p) and SU₃(Q_p) , while at the same time do not admit any representation which violates the Ramanujan Conjecture. This gives us Ramanujan complexes from PSL₃(F_p) and PSU₃(F_p) , as well as golden gates for PU(3).

Original language	English
Pages (from-to)	193-235
Number of pages	43
Journal	Geometric and Functional Analysis
Volume	32
Issue number	2
DOIs	https://doi.org/10.1007/s00039-022-00593-9
State	Published - Apr 2022

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© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

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title = "Ramanujan Complexes and Golden Gates in PU(3)",

abstract = "In a seminal series of papers from the 80{\textquoteright}s, Lubotzky, Phillips and Sarnak applied the Ramanujan–Petersson Conjecture for GL2 (Deligne{\textquoteright}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).",

author = "Shai Evra and Ori Parzanchevski",

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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).

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Ramanujan Complexes and Golden Gates in PU(3)

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