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Conformal Geometry and Dynamics

ISSN 1088-4173



Deformations of the modular group as a quasifuchsian correspondence

Author: Shaun Bullett
Journal: Conform. Geom. Dyn. 14 (2010), 296-321
MSC (2010): Primary 37F05; Secondary 37F30
Published electronically: November 18, 2010
MathSciNet review: 2738531
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Abstract: When viewed as a $(2:2)$ holomorphic correspondence on the Riemann sphere, the modular group $PSL_2({\mathbb Z})$ has a moduli space ${\mathcal Q}$ of non-trivial deformations for which the limit set remains a topological circle. This space is analogous to a Bers slice of the deformation space of a Fuchsian group as a Kleinian group, but there are certain differences. A Bers slice contains a single quasiconformal conjugacy class of Kleinian groups: we show that for an open dense set of parameter values in ${\mathcal Q}$ the correspondence belongs to a single quasi-conformal conjugacy class, but that at a countable set ${\mathcal C}$ of isolated parameter values it satisfies an additional critical relation. We classify these relations, propose ‘pleating coordinates’ for ${\mathcal Q}$, and investigate how the correspondence degenerates on the boundary of ${\mathcal Q}$. In particular, we show that there is a point on the boundary of ${\mathcal Q}$ where the correspondence degenerates into a mating between $PSL_2(\mathbb Z)$ and the quadratic polynomial $z \to z^2+1/4$. A key ingredient in our analysis is a bijection between ${\mathcal Q}\setminus {\mathcal C}$ and an intermediate cover between the moduli space of the space of non-critical grand orbits of the correspondence, and its universal cover, the corresponding Teichmüller space.

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Additional Information

Shaun Bullett
Affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

Received by editor(s): July 7, 2010
Published electronically: November 18, 2010
Additional Notes: The author thanks the Heilbronn Institute, University of Bristol, for its support during the writing of this paper, and the Fields institute, Toronto, for providing an ideal environment for early work in March 2006, on the ideas presented here.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.