** Topology of ends of nonpositively curved manifolds **

Joint work with T. Nguyen Pham.

I am interested in complete Riemannian manifolds with curvature in , and finite volume.

**Example**. Product of two hyperbolic surfaces. The end is homeomorphic to , with some extra structure: is made of two pieces.

More generally, for locally symmetric spaces of noncompact type, lifts of ends are homeomorphic to , with a wedge of spheres. This description goes back to Borel-Serre.

**1. Thick-thin decomposition **

Gromov-Schroeder: assume there are no arbitrarily small geodesic loops. Then the thin part is homeomorphic to , with a closed manifold.

The condition is necessary. Gromov gives an example of a nonpositively curved infinite type graph manifold of finite volume.

**Theorem 1 (Avramidi-Nguyen Pham)** * Under the same assumptions, any map of a polyhedron to the thin part of the universal cover can be homotoped within the thin part into a map to an -dimensional complex, . *

**Consequences**:

- If , each component of the thin part is aspherical and has locally free fundamental group.
- for all .
- .

**2. Proof **

Maximizing the angle under which two visual boundary points are seen gives Tits distance, and the corresponding path metric .

In the universal cover, the thin part is the set of points moved less than away by some deck transformation . Isometries are either hyperbolic (minimal displacement is achieved) or parabolic (infimal displacement is 0). Parabolic isometries have a nonempty fix-point set at infinity. At each point , the subgroup generated by isometries moving no more than is virtually nilpotent, hence virtually has a common fixed point at infinity. This allows to define a discontinuous projection to infinity. The point is to show that the image has dimension .

** 2.1. Busemann simplices **

If and are Busmeann functions, need not be a Busemann function again, but on each sphere, there is a unique point where it achieves its minimum, and tis point depends in a Lipschitz manner on . This defines an arc in Tits boundary, hence simplices . We claim that

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