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.

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