Cusps of Hilbert modular varieties

Details Cusps of Hilbert modular varieties Cusps of Hilbert modular varieties

2007-06-25

arxiv.org/abs/0706.3580v1

Math. Proc. Cambridge Philos. Soc. 2008 (144), 749-759

Mathematics

Geometric Topology

To appear in Mathematical Proceedings Cambridge Philosophical Society

Scientific paper

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 diffeomorphic 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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