2008/08/26 by Gilman, Jane, Keen, Linda · 1 citation
#30F40 #30F60 #32G15 #51M10 #57M99 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.0808.3488
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 \sl 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 \HH3 whether or not G is discrete. We show that there is a \sl core geodesic Ł 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 Ł; the second is that G is geometrically finite if and only if the axis of every non-parabolic palindromic word in G intersects Ł 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.