End Invariants for Sl(2,c) Characters of the One-holed Torus

We define and study the set E(ρ) of end invariants of a SL(2,C) character ρ of the one-holed torus T . We show that the set E(ρ) is the entire projective lamination space PL of T if and only if (i) ρ corresponds to the dihedral representation, or (ii) ρ is real and corresponds to a SU(2) representation; and that otherwise, E(ρ) is closed and has empty interior in PL . For real characters ρ, we give a complete classification of E(ρ), and show that E(ρ) has either 0, 1 or infinitely many elements, and in the last case, E(ρ) is either a Cantor subset of PL or is 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), E(ρ) is a Cantor subset of PL if it contains at least three elements.