On the Efficiency of Coxeter Groups

Abstract

If G is a finitely presented group and K is any (G,2)-complex (that is, a finite 2-complex with fundamental group G), then it is well known that X(K) ⩾ ν(G), where ν(G) = 1−rk H1G + dH2G. We define χ(G) to be min{χ(K): K a (G, 2)-complex}, and we say that G is efficient if χ(G)=ν(G). In this paper we give sufficient conditions for a Coxeter group to be efficient (Theorem 4.2). We also give examples of inefficient Coxeter groups (Theorem 5.1). In fact, we give an infinite family Gn(n = 2, 3, 4, …) of Coxeter groups such that χ(Gn)−ν(Gn) → ∞ as n → ∞. 1991 Mathematics Subject Classification 20F05, 20F55.