Constructs infinite families of (p^3+1, p+1)-regular Ramanujan Cayley bigraphs from PU_3(Q_p) lattices, analyzes RC cases, proves cutoff for non-backtracking walks, and gives applications to approximation and complexes.
Lp expander Complexes
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abstract
We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta function satisfies the Riemann hypothesis.
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2023 1verdicts
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Ramanujan Bigraphs
Constructs infinite families of (p^3+1, p+1)-regular Ramanujan Cayley bigraphs from PU_3(Q_p) lattices, analyzes RC cases, proves cutoff for non-backtracking walks, and gives applications to approximation and complexes.