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Cusps of Hilbert modular varieties

Published online by Cambridge University Press:  01 May 2008

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.

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
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]F, M.. Atiyah, Donnelly, H. and M, I.. Singer. Eta invariants, signature defects of cusps, and values of L–functions, Ann. of Math. (2) 118 (1983), no. 1, 131177.Google Scholar
[2]Borel, A. and Harish–Chandra, . Arithmetic subgroups of algebraic groups. Ann. of Math. (2) 75 (1962), 485535.CrossRefGoogle Scholar
[3]Cheeger, J. and Gromov, M.. Bounds on the von Neumann dimension of L 2–cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differential Geom. 21 (1985), no. 1, 134.CrossRefGoogle Scholar
[4]Farb, B. E. and Schwartz, R.. The large-scale geometry of hilbert modular groups. J. Differential Geom. 44 (1996), no. 3, 435–478.CrossRefGoogle Scholar
[5]C, G.. Hamrick and Royster, D. C.. Flat Riemannian manifolds are boundaries. Invent. Math. 66 (1982), no. 3, 405413.Google Scholar
[6]E, F.. Hirzebruch, P.. Hilbert modular surfaces. Enseign. Math. (2) 19 (1973), 183281.Google Scholar
[7]Long, D. D. and W, A.. Reid. On the geometric boundaries of hyperbolic 4–manifolds. Geom. Topol. 4 (2000), 171178 (electronic).CrossRefGoogle Scholar
[8]Long, D. D.. and Reid, A. W.. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol. 2 (2002), 285296(electronic).CrossRefGoogle Scholar
[9]McReynolds, D. B.. Peripheral separability and cusps of arithmetic hyperbolic orbifolds. Algebr. Geom. Topol. 4 (2004), 721755 (electronic).CrossRefGoogle Scholar
[10]Mostow, G. D.. Factor spaces of solvable groups. Ann. of Math. (2) 60 (1954), 127.CrossRefGoogle Scholar
[11]Petersen, K.. One-cusped congruence subgroups of PSL (2; Ok. Ph. d. Thesis, University of Texas (2005).CrossRefGoogle Scholar
[12]Rohlin, V. D.. A three-dimensional manifold is the boundary of a four-dimensional one. Doklady Akad. Nauk SSSR (N.S.) 81 (1951), 355357.Google Scholar
[13]Schwartz, R. E.. The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. (1995), no. 82, 133168 (1996).CrossRefGoogle Scholar
[14]Scott, P.. The geometries of 3–manifolds. Bull. London Math. Soc. 15 (1983), no. 5, 401487.CrossRefGoogle Scholar
[15]Shimizu, H.. On discontinuous groups operating on the product of the upper half planes. Ann. of Math. (2) 77 (1963), 3371.CrossRefGoogle Scholar
[16]van der Geer, G.. Hilbert modular surfaces. Ergeb. Math. Grenzgeb. (3), vol. 16 (1988).Google Scholar