The diagonal slice of Schottky space

Abstract

An irreducible representation of the free group on two generators X,Y into SL(2,C) is determined up to conjugation by the traces of X,Y and XY. We study the diagonal slice of representations for which X,Y and XY have equal trace. Using the three-fold symmetry and Keen-Series pleating rays we locate those groups which are free and discrete, in which case the resulting hyperbolic manifold is a genus-2 handlebody. We also compute the Bowditch set, consisting of those representations for which no primitive elements in the group generated by X,Y are parabolic or elliptic, and at most finitely many have trace with absolute value at most 2. In contrast to the quasifuchsian punctured torus groups originally studied by Bowditch, computer graphics show that this set is significantly different from the discreteness locus.