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Cusps of Hilbert modular varieties

Published online by Cambridge University Press:  01 May 2008

D. B. McREYNOLDS
D. B. McREYNOLDS*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A. email: [email protected]
*
†Supported in part by a V.I.G.R.E. graduate fellowship and Continuing Education fellowship.
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Abstract

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 749 - 759
Copyright
Copyright © Cambridge Philosophical Society 2008

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