We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS]). They are obtained as quotients of the buildings of type Ãd-1 associated with PGLd(F) where F is a local field of positive characteristic.
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1. Introduction A finite k-regular graph X is called a Ramanujan graph if for every eigenvalue A of the adjacency matrix A = Ax of X either A = ~=k or I)~1 < 2v/k - 1. This term was defined in [LPS] where some explicit constructions of such graphs were presented; see also [Mal], [Lul], [Mo]. These graphs were obtained as quotients of the k-regular tree T = Tk, for k = q+l, q a prime power, divided by the action * The authors were partially supported by grants from NSF and BSF (U.S.-Israel). ** Current address: Department of Mathematics, Bar-Ilan University, Ramat Gan 52900, Israel. Received September 1, 2003 and in revised form February 23, 2004