On the Poles of the Growth Series of Coxeter Groups

Abstract: We present an overview of the problems connected with the number of real roots of the growth series of Coxeter groups. Let (W, S) be a Coxeter system [B]. For any T ⊂ S, WT denotes a subgroup generated by T and is called a standard parabolic. The nerve NW of the Coxeter system (W, S) is the simplicial complex consisting of T ⊂ S such that the subgroup WT is finite. Definition. Assume that S is finite. A formal series W (t) = w∈W t ℓ (w), where ℓ is a length function with respect to S, is called a growth series ofW. The Euler Characteristic Conjecture. LetM be a compact aspherical manifold of dimen sion 2n, then The Euler characteristic satisfies (−1)nχ(M) ≥ 0. The construction of M. Davis [D] associates to (W, S) a contractible complex ΣW together with a geometric (cocompact with finite stabilizers) action of W. If the nerve NW of W is a simply connected generalized homology sphere (GHS) then ΣW is a manifold. By the Selberg Lemma [Sel] one finds a torsion free subgroup Γ of finite index in W. Since every element of W, with nontrivial fixpoint set, is conjugated into some WT of finite order [D], and thus torsion, the Euler Characteristic for Γ \ΣW reads Conjecture 1. Assume thatNW is a (simply connected) GHS of dimension (2n−1). Then (−1)