Ramanujan complexes, non-uniform quotients, and isospectrality.

摘要

We define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs (constructed in [LPS] and [Ma1]). 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.; Furthermore, Morgenstern [Mo2] generalized the notion of a Ramanujan graph to a "Ramanujan diagram", which is a weighted graph obtained by dividing the k-regular tree, B , by suitable non-uniform lattices of PGL2(F). Morgenstern showed that if F has positive characteristic and Gamma is a congruence subgroup of PGL2(F), then the diagram Gamma B is Ramanujan, that is the spectrum of the adjacency operator acting on L2(Gamma B ) is contained in the spectrum of L2( B ). We generalize this notion to complexes and show the opposite. If d ≥ 3, F = Fq (1/y)), R = Fq [y], and Gamma is a congruence subgroup of PGLd(R), then Gamma B is not Ramanujan. Here B = PGLd(F)/PGLd(O), where O is the ring of integers of F.; In addition, we show that for d ≥ 3, there are arithmetic lattices Gammai in PGLd(F) such that Gammai B are isospectral non-commensurable manifolds (in archimedean fields of characteristic zero), and complexes (in characteristic p). The constructions are based on arithmetic groups obtained from division algebras with the same ramification points but different invariants. In contrast, if Gamma 1 and Gamma2 are arithmetic lattices in PGL 2( R ) or in PGL2( C ) which give rise to isospectral manifolds, then it has been shown that Gamma1 and Gamma2 are always commensurable (after conjugation) [Re].