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Ramanujan complexes of type \(\widetilde A_d\). (English) Zbl 1087.05036
Summary: We define and construct Ramanujan complexes. These are simplicial complexes which are higher-dimensional analogues of Ramanujan graphs (constructed in [A. Lubotzky, R. Phillips and P. Sarnak, Combinatorica 8, 261–277 (1988; Zbl 0661.05035)]). They are obtained as quotients of the buildings of type \(\widetilde A_{d-1}\) associated with PGL\(_d(F)\) where \(F\) is a local field of positive characteristic.

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E42 Groups with a \(BN\)-pair; buildings
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