Degrees of maps between locally symmetric spaces

Abstract

Let $X$ be a locally symmetric space $\Gamma\backslash G/K$ where $G$ is a connected non-compact semisimple real Lie group with trivial centre, $K$ is a maximal compact subgroup of $G$, and $\Gamma\subset G$ is a torsion-free irreducible lattice in $G$. Let $Y=\Lambda\backslash H/L$ be another such space having the same dimension as $X$. Suppose that real rank of $G$ is at least $2$. We show that any $f:X\to Y$ is either null-homotopic or 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 $X$.