Discreteness criteria and the hyperbolic geometry of palindromes

Jane Gilman, Linda Keen

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We consider non-elementary representations of two generator free groups in PSL(2, C), not necessarily discrete or free, G =A, B. Aword 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 H3 whether or not G is discrete. We show that there is a core geodesic 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 nonnegative rationals and from the primitive elements to L; the second is that G is geometrically finite if and only if the axis of every non-parabolic palindromic word in G intersects 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.

Original languageEnglish (US)
Pages (from-to)76-90
Number of pages15
JournalConformal Geometry and Dynamics
Volume13
Issue number3
DOIs
StatePublished - Feb 17 2009

ASJC Scopus subject areas

  • Geometry and Topology

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