The diagonal slice of Schottky space

Abstract

An irreducible representation of the free group on two generators $X,Y$ into $SL\left(2,ℂ\right)$ is determined up to conjugation by the traces of $X,Y$ and $XY$. If the representation is faithful and discrete, the resulting manifold is in general a genus-$2$ handlebody. We study the diagonal slice of the representation variety in which $TrX=TrY=TrXY$. Using the symmetry, we are able to compute the Keen–Series pleating rays and thus fully determine the locus of faithful discrete representations. We also computationally determine the “Bowditch set” consisting of those parameter values for which no primitive elements in $\langle X,Y\rangle$ have traces in $\left[-2,2\right]$, and at most finitely many primitive elements have traces with absolute value at most $2$. The graphics make clear that this set is both strictly larger than, and significantly different from, the discreteness locus.