Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T05:18:11.453Z Has data issue: false hasContentIssue false
  • English
  • Français

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: dmcreyn@caltech.edu
*
†Supported in part by a V.I.G.R.E. graduate fellowship and Continuing Education fellowship.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 131–177.Google Scholar
[2]Borel, A. and Harish–Chandra, . Arithmetic subgroups of algebraic groups. Ann. of Math. (2) 75 (1962), 485–535.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, 1–34.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, 405–413.Google Scholar
[6]E, F.. Hirzebruch, P.. Hilbert modular surfaces. Enseign. Math. (2) 19 (1973), 183–281.Google Scholar
[7]Long, D. D. and W, A.. Reid. On the geometric boundaries of hyperbolic 4–manifolds. Geom. Topol. 4 (2000), 171–178 (electronic).CrossRefGoogle Scholar
[8]Long, D. D.. and Reid, A. W.. All flat manifolds are cusps of hyperbolic orbifolds. Algebr. Geom. Topol. 2 (2002), 285–296(electronic).CrossRefGoogle Scholar
[9]McReynolds, D. B.. Peripheral separability and cusps of arithmetic hyperbolic orbifolds. Algebr. Geom. Topol. 4 (2004), 721–755 (electronic).CrossRefGoogle Scholar
[10]Mostow, G. D.. Factor spaces of solvable groups. Ann. of Math. (2) 60 (1954), 1–27.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), 355–357.Google Scholar
[13]Schwartz, R. E.. The quasi-isometry classification of rank one lattices. Inst. Hautes Études Sci. Publ. Math. (1995), no. 82, 133–168 (1996).CrossRefGoogle Scholar
[14]Scott, P.. The geometries of 3–manifolds. Bull. London Math. Soc. 15 (1983), no. 5, 401–487.CrossRefGoogle Scholar
[15]Shimizu, H.. On discontinuous groups operating on the product of the upper half planes. Ann. of Math. (2) 77 (1963), 33–71.CrossRefGoogle Scholar
[16]van der Geer, G.. Hilbert modular surfaces. Ergeb. Math. Grenzgeb. (3), vol. 16 (1988).Google Scholar