Discreteness Criteria and the Hyperbolic Geometry of Palindromes

We consider non-elementary representations of two generator free groups in PSL(2,C), not necessarily discrete or free, G = 〈A,B〉. 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 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.