Discreteness criteria and the hyperbolic geometry of palindromes
Authors:
Jane Gilman and Linda Keen
Journal:
Conform. Geom. Dyn. 13 (2009), 7690
MSC (2000):
Primary 30F10, 30F35, 30F40; Secondary 14H30, 22E40
Published electronically:
February 17, 2009
MathSciNet review:
2476657
Fulltext PDF Free Access
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Abstract: We consider nonelementary representations of two generator free groups in , not necessarily discrete or free, . A word in and , , 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 in whether or not is discrete. We show that there is a core geodesic in the convex hull of the limit set of and use it to prove three results: the first is that there are welldefined maps from the nonnegative rationals and from the primitive elements to ; the second is that is geometrically finite if and only if the axis of every nonparabolic palindromic word in intersects 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:
http://dx.doi.org/10.1090/S108841730900191X
PII:
S 10884173(09)00191X
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 PSCCUNY
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
