A Moore bound for simplicial complexes

Abstract

Let X be a d-dimensional simplicial complex with N faces of dimension d − 1. Suppose that every (d − 1)-face of X is contained in at least k d + 2 faces of X, of dimension d. Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at mostlog k−d Nwhose d-dimensional homology Hd(B) is non-zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor. This implies the classical Moore bound. Theorem A. g(G) < 2 logk−1 n +2 . Let dG(u, v) be the distance between the vertices u and v in the graph metric, and let Br(v )= {u ∈ V : dG(u, v) r} denote the ball of radius r around v. Define the acyclicity radius rv(G )o fG at the vertex v to be the maximal r such that the induced graph G(Br(v)) is acyclic. Let r(G) = minv∈V rv(G); then r(G )= � g(G)/2 �− 1. The asymptotic version of Moore's bound is equivalent to the following statement. Theorem A1. If δ(G )= k 3, then for every v ∈ V rv(G) � log k−1 n� . (1.2)