Explicit Constructions of Ramanujan Complexes
نویسندگان
چکیده
In [LSV] we defined and proved the existence of Ramanujan complexes, see also [B1], [CSZ] and [Li]. The goal of this paper is to present an explicit construction of such complexes. Our work is based on the lattice constructed by Cartwright and Steger [CS]. This remarkable discrete subgroup Γ of PGLd(F ), when F is a local field of positive characteristic, acts simply transitively on the vertices of the Bruhat-Tits building Bd(F ), associated with PGLd(F ). By choosing suitable congruence subgroups of Γ, we are able to present the 1-skeleton of the corresponding finite quotients of Bd(F ) as Cayley graphs of explicit finite groups, with specific sets of generators. The simplicial complex structure is then defined by means of these generators. Let [ k]q denote the number of subspaces of dimension k of F d q . Theorem 1.1. Let q be a prime power, d ≥ 2, e ≥ 1 (e > 1 if q = 2). Then, the group G = PGLd(Fqe) has an (explicit) set S of [d1]q+[ d 2]q+ · · ·+ [ d d−1]q generators, such that the Cayley complex of G with respect to S is a Ramanujan complex, covered by Bd(F ), when F = Fq((y)).
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