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Conformal Geometry and Dynamics

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Discreteness criteria and the hyperbolic geometry of palindromes


Authors: Jane Gilman and Linda Keen
Journal: Conform. Geom. Dyn. 13 (2009), 76-90
MSC (2000): Primary 30F10, 30F35, 30F40; Secondary 14H30, 22E40
DOI: https://doi.org/10.1090/S1088-4173-09-00191-X
Published electronically: February 17, 2009
MathSciNet review: 2476657
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Abstract: We consider non-elementary representations of two generator free groups in $ PSL(2,\mathbb{C})$, not necessarily discrete or free, $ G = \langle A, B \rangle$. A word in $ A$ and $ B$, $ W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is primitive if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $ G$ in $ {\mathbb{H}}^3$ whether or not $ G$ is discrete. We show that there is a core geodesic $ {\mathbf{L}}$ in the convex hull of the limit set of $ G$ and use it to prove three results: the first is that there are well-defined maps from the non-negative rationals and from the primitive elements to $ {\mathbf{L}}$; the second is that $ G$ is geometrically finite if and only if the axis of every non-parabolic palindromic word in $ G$ intersects $ {\mathbf{L}}$ in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.


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Additional Information

Jane Gilman
Affiliation: Department of Mathematics, Rutgers University, Newark, New Jersey 07079
Email: gilman@rutgers.edu

Linda Keen
Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York, New York 10468
Email: Linda.keen@lehman.cuny.edu

DOI: https://doi.org/10.1090/S1088-4173-09-00191-X
Received by editor(s): December 29, 2008
Published electronically: February 17, 2009
Additional Notes: The first author was supported in part by the Rutgers Research Council and Yale University
The second author was supported in part by the PSC-CUNY
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.