End Invariants for $\SL(2,C)$ characters of the one-holed torus

Abstract

We define and study the set ${\mathcal E}(\rho)$ of end invariants of a $\SL(2,C)$ character $\rho$ of the one-holed torus $T$. We show that the set ${\mathcal E}(\rho)$ is the entire projective lamination space $\mathscr{PL}$ of $T$ if and only if (i) $\rho$ corresponds to the dihedral representation, or (ii) $\rho$ is real and corresponds to a SU(2) representation; and that otherwise, ${\mathcal E}(\rho)$ is closed and has empty interior in $\mathscr{PL}$. For real characters $\rho$, we give a complete classification of ${\mathcal E}(\rho)$, and show that ${\mathcal E}(\rho)$ has either 0, 1 or infinitely many elements, and in the last case, ${\mathcal E}(\rho)$ is either a Cantor subset of $\mathscr{PL}$ or is $\mathscr{PL}$ itself. We also give a similar classification for "imaginary" characters where the trace of the commutator is less than 2. Finally, we show that for discrete characters (not corresponding to dihedral or SU(2) representations), ${\mathcal E}(\rho)$ is a Cantor subset of $\mathscr{PL}$ if it contains at least three elements.