On polyhedral retracts and compactifications of locally symmetric spaces

The results in this paper are based on a previously constructed exhaustion of a locally symmetric space V=(gamma)\X by Riemannian polyhedra, i.e., compact submanifolds with corners: V=Ux(greater than)0 v(s) We show that the interior of every polyhedron V(s) is homeomorphic to V. The universal covering space X(s) of V(s) is quasi-isometric to the discrete group (gamma). It can be written as the complement of a (gamma)-invariant union of horoballs in X (which in general have intersections giving rise to the corners). This yields exponential isoperimetric inequalities for (GAMMA)(approximately equal)(pi)1 (V(s)). We also discuss the relation of this compactification of V with the Borel– Serre compactification.