Ramanujan Complexes of Type $\widetilde{A_d}$

Abstract

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs. They are obtained as quotients of the buildings of type ${\tilde{A}}_{d-1}$ associated with ${\mathrm{PGL}}_d(F)$ where $F$ is a local field of positive characteristic.

32 pages