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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Received: May 25, 1994; revised version: September 4, 1996
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Saper, L. Tilings and finite energy retractions of locally symmetric spaces. Comment. Math. Helv. 72, 167–202 (1997). https://doi.org/10.1007/PL00000369
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DOI: https://doi.org/10.1007/PL00000369