The fundamental group at infinity

Abstract

Let G be a finitely presented infinite group which is semistable at infinity, let X be a finite complex whose fundamental group is G, and let ω be a base ray in the universal covering space X ̃ . The fundamental group at ∞ of G is the topological group π 1 e( X, ω) ≡ lim {π 1 ( X − L)∣L ⊂ X is compact} . We prove the following analogue of Hopf's theorem on ends: π 1 e( X ̃ ,ω) is trivial, or is infinite cyclic, or is freely generated by a non-discrete pointed compact metric space; or else the natural representation of G in the outer automorphisms of π 1 e( X ̃ ,ω) has torsion kernel . A related manifold result is: Let G be torsion free (not necessarily finitely presented) and act as covering transformations on a connected manifold M so that the quotient of M by any infinite cyclic subgroup is non-compact; if M is semistable at ∞ then the natural representation of G in the mapping class group of M is faithful. The latter theorem has applications in 3-manifold topology.