Math @ Duke

Publications [#244108] of Leslie Saper
Papers Published
 Saper, L, Tilings and finite energy retractions of locally symmetric spaces,
Commentarii Mathematici Helvetici, vol. 72 no. 2
(January, 1997),
pp. 167201, Research Institute for Mathematical Sciences, Kyoto University [MR99a:22019], [doi]
(last updated on 2022/01/16)
Abstract: Let Γ\X̄ be the BorelSerre 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 ArthurLanglands 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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