Tilings and finite energy retractions of locally symmetric spaces

Let $ \Gamma \backslash \overline{X} $ be the Borel-Serre compactifiction of an arithmetic quotient $ \Gamma \backslash X $ of a symmetric space of noncompact type. We construct natural tilings $ \Gamma \backslash \overline{X} = \coprod _P \Gamma \backslash \overline{X}_P $ (depending on a parameter b) which generalize the Arthur-Langlands partition of $ \Gamma \backslash X $. This is applied to yield a natural piecewise analytic deformation retraction of $ \Gamma \backslash \overline{X} $ onto a compact submanifold with corners $ \Gamma \backslash X _0 \subset \Gamma \backslash X $. In fact, we prove that $ \Gamma \backslash X _0 $ is a realization (under a natural piecewise analytic diffeomorphism) of $ \Gamma \backslash \overline{X} $ inside the interior $ \Gamma \backslash X $. For application to the theory of harmonic maps and geometric rigidity, we prove this retraction and diffeomorphism have finite energy except for a few low ranks examples. We also use tilings to give an explicit description of a cofinal family of neighborhoods of a face of $ \Gamma \backslash \overline{X} $, and study the dependance of tilings on the parameter b and the degeneration of tilings.