  Every lattice H in a connected semi-simple Lie group G acts properly
discontinuously by isometries on the contractible manifold G/K (K a maximal
compact subgroup of G). We prove that if H acts on a contractible manifold W
and if either
  1) the action is properly discontinuous, or
  2) W is equipped with a complete Riemannian metric, the action is by
isometries and with unbounded orbits, G is simple with finite center and rank
>1, then dim W is not less than dim G/K.
