Tilings and finite energy retractions of locally symmetric spaces
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.
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- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics
Citation
Published In
DOI
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics