RAMANUJAN COMPLEXES OF TYPE Ad*
نویسندگان
چکیده
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 Ad-1 associated with PGLd(F) where F is a local field of positive characteristic. 1. I n t r o d u c t i o n A finite k-regular graph X is called a Ramanujan graph if for every eigenvalue A of the adjacency matr ix 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
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