An analog of the rational Tits building in nonpositive curvature

This talk is about nonpositively curved geometry, no knowledge of
Tits buildings is required (or will be given).

Locally symmetric manifolds of noncompact type form an interesting class of
nonpositively curved manifolds. The topology of the end of an arithmetic
locally symmetric space is controlled by the rational Tits building, which
is homotopically a wedge of spheres of dimension q-1, where q is the Q-rank
of the locally symmetric space. In general, q is less than or equal to half
the dimension of the locally symmetric space. We show that this is not an
arithmetic coincidence by building an analog of the rational Tits building
for general noncompact, finite volume, complete, bounded nonpositively

curved n-manifolds with no arbitrarily small geodesic loops (so that M is
tame by a theorem of Gromov-Schroeder). We use this is show that any
polyhedron in the thin part of M that lifts to the universal cover
\tilde{M} can be homotoped within the thin part of M to one with dimension
less than or equal to (n/2 - 1). I will describe how this is done.