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Cusps of Hilbert modular varieties

McReynolds, D. B. (2008) Cusps of Hilbert modular varieties. Mathematical Proceedings of the Cambridge Philosophical Society, 144 (3). pp. 749-759. ISSN 0305-0041. doi:10.1017/S0305004107001004.

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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.

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Additional Information:© 2008 Cambridge Philosophical Society. Received 23 November 2006; revised 26 June 2007. I would like to thank my advisor Alan Reid for all his help. In addition, I would like to thank Richard Schwartz for suggesting Hilbert modular varieties as a family of examples for which the techniques developed in [9] might be applied and for carefully reading an early draft of this paper. Supported in part by a V.I.G.R.E. graduate fellowship and Continuing Education fellowship.
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Vertical InteGration of Research and Education (V.I.G.R.E.)UNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20090519-100547212
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14265
Deposited By: Tony Diaz
Deposited On:01 Jun 2009 18:01
Last Modified:08 Nov 2021 23:10

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