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Cusp closing in rank one symmetric spaces


Hummel, C; Schroeder, V (1996). Cusp closing in rank one symmetric spaces. Inventiones Mathematicae, 123(2):283-307.

Abstract

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic of Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel [resp. Siu-Yau] compactification of finite volume quotients of the complex hyperbolic space.

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic of Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel [resp. Siu-Yau] compactification of finite volume quotients of the complex hyperbolic space.

Citations

9 citations in Web of Science®
9 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1996
Deposited On:29 Nov 2010 16:28
Last Modified:05 Apr 2016 13:27
Publisher:Springer
ISSN:0020-9910
Free access at:Related URL. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s002220050027
Related URLs:http://www.digizeitschriften.de/dms/img/?PPN=PPN356556735_0123&DMDID=dmdlog19
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0860.53025

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