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Published May 2008 | Published + Erratum
Journal Article Open

Cusps of Hilbert modular varieties

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.

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.

Attached Files

Published - MCRmpcps08.pdf

Erratum - MCRmpcps08corr.pdf

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August 19, 2023
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