THE FUNDAMENTAL GROUP AT INFINITY

LET G be a finitely presented infinite group which is semistable at in finity, let X be a finite complex whose fundamental group is G, and le t omega be a base ray in the universal covering space (X) over tilde. The fundamental group at oo of G is the topological group pi(1)(e)((X) over tilde, omega) = lim {pi(1)((X) over tilde - L)\L subset of (X) o ver tilde is compact}. We prove the following analogue of Hopfs theore m on ends: pi(1)(e)((X) over tilde, omega) is trivial, or is infinite cyclic, or is freely generated by a non-discrete pointed compact metri c space; or else the natural representation of G in the outer automorp hisms of pi(1)(e)((X) over tilde, omega) has torsion kernel. A related manifold result is: Let G be torsion free (not necessarily finitely p resented) and act as covering transformations on a connected manifold M so that the quotient of M by any infinite cyclic subgroup is non-com pact; if M is semistable at infinity then the natural representation o f G in the mapping class group of M is faithful. The latter theorem ha s applications in 3-manifold topology. Copyright (C) 1996 Elsevier Sci ence Ltd