A refinement of the simple connectivity at infinity for groups

Abstract

We give another proof for a result of Brick ([2]) stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of $\pi^{\infty}_{1}$ for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for fundamental groups of geometric 3-manifolds.