On non-uniform Ramanujan complexes

Abstract

where Ax is the adjacency matrix of the graph X. Such graphs were constructed there as quotients of the Bruhat-Tits tree associated to G = PGL2(QP), modulo congruence subgroups of uniform (i.e. cocompact) lattices of G, where Qp is the field of the p-adic integers. More examples were given by Morgenstern [Ml] by replacing Qp by a local field F of characteristic p > 0, such as ¥q((t)). One significant difference is that unlike PGL2(QP), PGL»2(Fg ((£))) has a nonuniform lattice A = PGIj2(Fg[l/t]), which is of finite covolume out not cocompact. For congruence subgroups of A, the quotient graphs are infinite, but the edges and vertices come with weights, so that the total weight is finite. Morgenstern [M2] went ahead to define Ramanujan diagrams as such weighted objects satisfying Equation (1.1), where this time Ax is the 'weighted adjacency matrix' (see Section 2 for more details) . He constructed explicit examples and even showed a surprising application of them to the theory of networks [M3] . Recently, the concept of Ramanujan graphs was generalized to higher dimensional complexes [Li, LSV1, Sa] (see Remark 4.5.(2)) by replacing PGL/2(F) by PGLd(F) for d > 2. This time one requires bounds on the simultaneous spectrum of d 1 'Hecke operators', which are 'colored adjacency matrices'; see Section 3 below.