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Ramanujan Complexes and Golden Gates in PU(3)

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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).

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Notes

  1. Recall that the Schreier graph Sch(X, S) has vertex set X, and the edges \(\{(x\rightarrow sx)|x\in X,s\in S\}\).

  2. For \(N=3,5\) this holds unconditionally, hence so does Theorem 1.2 with \(q=3,5\) (see Prop. 5.12)

  3. A warning is in order: for trees, a symmetric set which takes a vertex once to each neighbor always generates a group which acts simply-transitively. In higher dimensions this is far from true!

  4. By a Cayley complex we mean a clique complex of a Cayley graph.

  5. In the case of a general Hermitian form H, one should take \(g^{\#}=H(g^{*})^{-1}H^{-1}\) correspondingly.

  6. In contrast, \(\left( {\begin{matrix}1 &{} 2\\ 2 &{} 1 \end{matrix}}\right) ^{*}\left( {\begin{matrix}1\\ &{} -1 \end{matrix}}\right) \left( {\begin{matrix}1 &{} 2\\ 2 &{} 1 \end{matrix}}\right) =\left( {\begin{matrix}-3\\ &{} 3 \end{matrix}}\right) \) implies that \(PGU_{2}\left( {\mathbb {R}},\left( {\begin{matrix}1\\ &{} -1 \end{matrix}}\right) \right) \ne PU_{2}\left( {\mathbb {R}},\left( {\begin{matrix}1\\ &{} -1 \end{matrix}}\right) \right) \).

  7. The \(\Lambda \) we shall eventually consider are \(\Gamma _{p}\) for Theorem 1.1(1), \(\Lambda _{p}\) for 1.2(1), \(\Lambda _{p}(q)\) for 1.2(2,3), and \(\Lambda _{p}[q]\) for 1.1(2,3).

  8. In the language of Hecke algebras, \(\alpha _{S}\) corresponds to the element \(\mathbbm {1}_{K_{p}SK_{p}}\in C_{c}\left( K_{p}\backslash G_{p}/K_{p}\right) \).

  9. For example, \(K=\{g\in G(\hat{{\mathbb {Z}}})|g\equiv I\left( N\right) \}\) with \(p\not \mid N\), for which \(G({\mathbb {Q}})\cap K^{p} =\Gamma _{p}(N)\).

  10. Conjecturally, this also holds at the ramified primes—see Remark 5.8.

  11. The same holds for \(p\equiv 3(4)\) if one takes Definition 2.4 for graphs as well.

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Acknowledgements

The authors are grateful to Peter Sarnak and Alex Lubotzky for their guidance, and thank Frank Calegari, Ana Caraiani, Pierre Deligne, Brooke Feigon, Yuval Flicker, Simon Marshall and the anonymous referees for helpful comments and discussions. S.E. was supported by ERC Grant 692854 and NSF Grant DMS-1638352, and O.P. by ISF Grant 2990/21.

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    Shai Evra & Ori Parzanchevski

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Evra, S., Parzanchevski, O. Ramanujan Complexes and Golden Gates in PU(3). Geom. Funct. Anal. 32, 193–235 (2022). https://doi.org/10.1007/s00039-022-00593-9

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