On Homotopy Properties of Certain Coxeter Group Boundaries

There is a canonical homomorphism ψ : π1(bdyX)→ π∞ 1 (X) from the fundamental group of the visual boundary, here denoted by bdy X, of any non-positively curved geodesic space X into its fundamental group at infinity. In this setting, the latter group coincides with the first shape homotopy group of the visual boundary: π∞ 1 (X) ≡ π̌1(bdy X). The induced homomorphism φ : π1(bdy X)→ π̌1(bdy X) provides a way to study the relationship between these groups. We present a class Z of compacta, so-called trees of manifolds, for which we can show that the homomorphisms φ : π1(Z)→ π̌1(Z) (Z ∈ Z) are injective. This class Z includes the visual boundaries Z = bdy X which arise from right-angled Coxeter groups whose nerves are closed PL-manifolds. In particular, it includes the visual boundaries of those Coxeter groups which act on Davis’ exotic open contractible manifolds [2]. 1. The first shape homotopy group of a metric compactum We recall the definition of the first shape homotopy group of a pointed compact metric space (Z, z0). Choose an inverse sequence (Z1, z1) f2,1 ←− (Z2, z2) f3,2 ←− (Z3, z3) f4,3 ←− · · · of pointed compact polyhedra such that (Z, z0) = lim ←− ((Zi, zi), fi+1,i). The first shape homotopy group of Z based at z0 is then given by π̌1(Z, z0) = lim ←− ( π1(Z1, z1) f2,1# ←− π1(Z2, z2) f3,2# ←− π1(Z3, z3) f4,3# ←− · · · ) . This definition of π̌1(Z, z0) does not depend on the choice of the sequence ((Zi, zi), fi+1,i) [8]. Let pi : (Z, z0) → (Zi, zi) be the projections of the limit (Z, z0) into its inverse sequence ((Zi, zi), fi+1,i) such that pi = fi+1,i ◦ pi+1 for all i. Since the maps pi induce homomorphisms Research of the first author supported in part by the Faculty Internal Grants Program of Ball State University. Research of the second author supported in part by NSF Grant DMS-0072786.