Spectrum and combinatorics of two-dimensional Ramanujan complexes
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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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- [Fir16]U. A. First, The Ramanujan property for simplicial complexes, preprint, arXiv:1605.02664.Google Scholar
- [Gar73]H. Garland, p-adic curvature and the cohomology of discrete subgroups of p-adic groups, Annals of Mathematics 97 (1973), 375–423.Google Scholar
- [GS14]A. Gundert and M. Szedlák, Higher dimensional Cheeger inequalities, in Computational Geometry SoCG’14, ACM, New York, 2014, pp. 181–188.Google Scholar
- [Hof70]A. J. Hoffman, On eigenvalues and colorings of graphs, in Graph Theory and its Applications (Proc. Advanced Sem., Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1969), Academic Press, New York, 1970, pp. 79–91.Google Scholar
- [Sar90]P. Sarnak, Some Applications of Modular Forms, Cambridge Tracts in Mathematics, Vol. 99, Cambridge University Press, Cambridge, 1990.Google Scholar