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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.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 20E42 Groups with a $$BN$$-pair; buildings
##### Keywords:
Ramanujan graphs; buildings
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