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.