# Tilings and finite energy retractions of locally symmetric spaces

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## 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.

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