Let be the Borel-Serre compactifiction of an arithmetic quotient of a symmetric space of noncompact type. We construct natural tilings (depending on a parameter b) which generalize the Arthur-Langlands partition of . This is applied to yield a natural piecewise analytic deformation retraction of onto a compact submanifold with corners . In fact, we prove that is a realization (under a natural piecewise analytic diffeomorphism) of inside the interior . 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 , and study the dependance of tilings on the parameter b and the degeneration of tilings.
Cite this article
L. Saper, Tilings and finite energy retractions of locally symmetric spaces. Comment. Math. Helv. 72 (1997), no. 2, pp. 167–202DOI 10.1007/PL00000369