The topology of the end of a finite volume, nonpositively curved, locally symmetric manifold M is

well understood. In particular, the region E of the universal cover lying over the end of M is homotopy

equivalent to an infinite wedge of (q-1)-spheres, where q (called the rational rank when M is arithmetic)

is at most (dim M)/2. In this talk I will discuss the topology of E for more general finite volume,

nonpositively curved manifolds. It turns out that if M has no arbitrarily small closed geodesic loops,

then any finite polyhedron in E can be deformed (in E) to have image of dimension < dim M/2.

The deformation uses simplices that are constructed using convex combinations of Busemann

functions (Busemann simplices). It shows that the upper bound on the dimension in which topology

occurs in E is a general phenomenon. On the other hand, the concentration in a single dimension

is not. In contrast to the locally symmetric case, there are examples where E has homology

simultaneously in all the dimensions 0,1,2, ... ,(dim M/2)-1.

Joint work with Tam Nguyen Phan.

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