Tilings and finite energy retractions of locally symmetric spaces

Published

Journal Article

Let Γ\X̄ be the Borel-Serre compactification of an arithmetic quotient Γ\X of a symmetric space of noncompact type. We construct natural tilings Γ\X̄ = ∐P Γ\X̄P (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\X̄ onto a compact submanifold with corners Γ\X0 ⊂ Γ\X. In fact, we prove that Γ\X0 is a realization (under a natural piecewise analytic diffeomorphism) of Γ\X̄ 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 rank examples. We also use tilings to give an explicit description of a cofinal family of neighborhoods of a face of Γ\X̄, and study the dependance of tilings on the parameter b and the degeneration of tilings.

Duke Authors

Cited Authors

  • Saper, L

Published Date

  • December 1, 1997

Published In

Volume / Issue

  • 72 / 2

Start / End Page

  • 167 - 201

International Standard Serial Number (ISSN)

  • 0010-2571

Citation Source

  • Scopus