Ramanujan graphs have extremal spectral properties, which imply a remarkable combinatorial behavior. In this paper we compute the high dimensional Hodge–Laplace spectrum of Ramanujan triangle complexes, and show that it implies a combinatorial expansion property, and a pseudorandomness result. For this purpose we prove a Cheeger-type inequality and a mixing lemma of independent interest.
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© 2019, The Hebrew University of Jerusalem.