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

• Autores: Ser Peow Tan, Yan Loi Wong, Ying-Ying Zhang
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 130, Nº 2, 2008, págs. 385-412
• Idioma: inglés
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• Resumen

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