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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Discreteness criteria and the hyperbolic geometry of palindromes
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by Jane Gilman and Linda Keen PDF
Conform. Geom. Dyn. 13 (2009), 76-90 Request permission

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
  • MR Author ID: 190609
  • Email: gilman@rutgers.edu
  • Linda Keen
  • Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York, New York 10468
  • MR Author ID: 99725
  • Email: Linda.keen@lehman.cuny.edu
  • 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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2476657