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Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds

Dunfield, Nathan M. and Ramakrishnan, Dinakar (2010) Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds. American Journal of Mathematics, 132 (1). pp. 53-97. ISSN 0002-9327. http://resolver.caltech.edu/CaltechAUTHORS:20100218-101507964

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Abstract

We exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston’s Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover, and we give a more precise quantitative lower bound. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried’s dynamical characterization of the fibered faces. The origin of the basic fibration M → S^1 is the modular elliptic curve E = X_0(49), which admits multiplication by the ring of integers of Q[√(−7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K = Q[√(−3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of O^∗_D of level 7. To analyze the topological properties of M, we use a new practical method for computing the Thurston norm, which is of independent interest. We also give a noncompact finite-volume hyperbolic 3-manifold with the same properties by using a direct topological argument.


Item Type:Article
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http://dx.doi.org/10.1353/ajm.0.0098DOIUNSPECIFIED
http://muse.jhu.edu/journals/american_journal_of_mathematics/summary/v132/132.1.dunfield.htmlPublisherUNSPECIFIED
Additional Information:© 2010 Johns Hopkins University Press. Manuscript received January 7, 2008; revised October 14, 2008. Research of both authors supported in part by the NSF; research of the first author supported in part by the Sloan Foundation.
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Sloan FoundationUNSPECIFIED
Subject Keywords:Three-manifolds (Topology)
Record Number:CaltechAUTHORS:20100218-101507964
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100218-101507964
Official Citation:Nathan M. Dunfield and Dinakar Ramakrishnan. "Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds." American Journal of Mathematics 132.1 (2010): 53-97. Project MUSE. [Library name], [City], [State abbreviation]. 26 Jan. 2010 <http://muse.jhu.edu/>.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17518
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:19 Feb 2010 18:38
Last Modified:26 Dec 2012 11:47

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