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 $\mathcal{PL}$ 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 $\mathcal{PL}$. 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 $\mathcal{PL}$ or is $\mathcal{PL}$ 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 $\mathcal{PL}$ if it contains at least three elements.

24 pages, 6 figures

References in corpus (2)

• The Modular Group Action on Real SL(2)-characters of a One-Holed Torus
• An exposition of results of Fricke