End invariants for SL(2,ℂ) characters of the one-holed torus

Abstract

We define and study the set ε(p) of end invariants of an SL(2, ℂ) character p of the one-holed torus T. We show that the set ε(p) is the entire projective lamination space ℘ℒ of T if and only if p corresponds to the dihedral representation or p is real and corresponds to an SU(2) representation; and that otherwise, ε(p) is closed and has empty interior in ℘ℒ. For real characters p, we give a complete classification of ε(p), and show that ε(p) has either 0, 1 or infinitely many elements, and in the last case, ε(p) is either a Cantor subset of ℘ℒ or is ℘ℒ 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 characters with discrete simple length spectrum (not corresponding to dihedral or SU(2) representations), ε(p) is a Cantor subset of ℘ℒ if it contains at least three elements. © 2008 by The Johns Hopkins University Press.