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On the topological classification of the floors of certain Hilbert fundamental domains


Author: Michael H. Hall
Journal: Proc. Amer. Math. Soc. 28 (1971), 67-70
MSC: Primary 10.21
DOI: https://doi.org/10.1090/S0002-9939-1971-0271029-3
MathSciNet review: 0271029
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Abstract: Associated to the field $ Q({k^{1/2}})$ ($ k$ a positive square free integer greater than one), there is a group of transformations of the product of two upper half planes which is analogous to the Hilbert modular group. This group has been shown to have a fundamental domain bounded by a finite number of hypersurfaces. Of particular interest is a subspace of the domain known as the ``floor.'' This floor is a quotient space of a fiber bundle over the circle which is determined by the field $ Q({k^{1/2}})$. The principal result of this paper is that, conversely, the topological type (indeed the homotopy type) of this fiber bundle determines the field $ Q({k^{1/2}})$ which gives rise to it. This is accomplished by computing the homology groups of the fiber space and showing that the integer $ k$ can be determined from these groups.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271029-3
Keywords: Hilbert modular group, fundamental domain, homology groups, quadratic fields
Article copyright: © Copyright 1971 American Mathematical Society

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