# Tilings and finite energy retractions of locally symmetric spaces

### L. Saper

Duke University, Durham, USA

## Abstract

Let $Γ\X$ be the Borel-Serre compactifiction of an arithmetic quotient $Γ\X$ of a symmetric space of noncompact type. We construct natural tilings $Γ\X=∐_{P}Γ\X_{P}$ (depending on a parameter b) which generalize the Arthur-Langlands partition of $Γ\X$ . This is applied to yield a natural piecewise analytic deformation retraction of $Γ\X$ onto a compact submanifold with corners $Γ\X_{0}⊂Γ\X$ . In fact, we prove that $Γ\X_{0}$ is a realization (under a natural piecewise analytic diffeomorphism) of $Γ\X$ inside the interior $Γ\X$ . 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 $Γ\X$ , 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. 167–202

DOI 10.1007/PL00000369