Abstract: The structure of ends of a finite volume, nonpositively curved, locally symmetric manifolds M is very well understood. By work of Borel and Serre, the structure of ends is captured by the rational Tits building. This is a simplicial complex built out of the algebra of the locally symmetric space which turns out to have dimension < dim M/2. In this talk, I will explain aspects of the locally symmetric situation that are true for more general finite volume nonpositively curved manifolds satisfying a mild tameness assumption (there are no arbitrarily small closed geodesic loops). The main result is a half-dimensional collapse phenomenon: the homology of the thin part of the universal cover vanishes in dimension greater or equal to dim M/2. One application is that any complex X homotopy eqiuvalent to M has dimension >= dim M/2. Another application is that the group cohomology with group ring coefficients of the fundamental group of M vanishes in low dimensions (< dim M/2). Joint work with Tam Nguyen Phan.
Topology of ends of nonpositively curved manifolds