Non-aspherical ends and nonpositive curvature

TL;DR

This paper investigates the topological restrictions on manifolds bounding nonpositively curved spaces, showing conditions under which the boundary must be aspherical and incompressible, based on the fundamental group's properties.

Contribution

It establishes new topological constraints for manifolds bounding nonpositively curved manifolds, especially relating to fundamental groups and their algebraic properties.

Findings

B must be aspherical and incompressible under certain conditions.

Fundamental group G's properties impose topological restrictions.

Results apply to manifolds with finite volume and specific group structures.

Abstract

We obtain restrictions on the topology of a closed connected manifold B that bounds a (possibly noncompact) manifold whose interior V admits a complete Riemannian metric of nonpositive sectional curvature. If G denotes the fundamental group of B, then a sample result is that B must be aspherical and incompressible if one of the following is true: (1) V has finite volume and G is virtually nilpotent, (2) G is virtually nilpotent and has no proper torsion-free quotients, (3) G is isomorphic to a uniform, irreducible lattice of real rank > 1.