Tilings and finite energy retractions of locally symmetric spaces

Abstract

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.