Degrees of maps between locally symmetric spaces

Let X be a locally symmetric space Gamma\G/K where G is a connected non-compact semisimple real Lie group with trivial centre, K is a maximal compact subgroup of G, and Gamma subset of G is a torsion-free irreducible lattice in G. Let Y = Lambda\H/L be another such space having the same dimension as X. Suppose that real rank of G is at least 2. We show that any f : X -> Y is either null-homotopic or it is homotopic to a covering projection of degree an integer that depends only on Gamma and Lambda. As a corollary we obtain that the set [X, Y] of homotopy classes of maps from X to Y is finite. We obtain results on the (non-)existence of orientation reversing diffeomorphisms on X as well as the fixed point property for X. (C) 2015 Elsevier Masson SAS. All rights reserved.