On the Efficiency of Coxeter Groups

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 H 1 G + dH 2 G . 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 G n ( n = 2, 3, 4, …) of Coxeter groups such that χ( G n )−ν( G n ) → ∞ as n → ∞. 1991 Mathematics Subject Classification 20F05, 20F55.