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

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dc.creatorTan, Ser Peow
dc.creatorWong, Yan Loi
dc.creatorZhang, Ying
dc.date2005-11-25
dc.date.accessioned2026-07-07T09:40:27Z
dc.date.available2026-07-07T09:40:27Z
dc.descriptionWe define and study the set ${\mathcal E}(ρ)$ of end invariants of a $\SL(2,C)$ character $ρ$ of the one-holed torus $T$. We show that the set ${\mathcal E}(ρ)$ is the entire projective lamination space $\mathscr{PL}$ 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, ${\mathcal E}(ρ)$ is closed and has empty interior in $\mathscr{PL}$. For real characters $ρ$, we give a complete classification of ${\mathcal E}(ρ)$, and show that ${\mathcal E}(ρ)$ has either 0, 1 or infinitely many elements, and in the last case, ${\mathcal E}(ρ)$ 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), ${\mathcal E}(ρ)$ is a Cantor subset of $\mathscr{PL}$ if it contains at least three elements.
dc.description24 pages, 6 figures
dc.identifierhttps://arxiv.org/abs/math/0511621
dc.identifierhttp://arxiv.org/abs/math/0511621
dc.identifierAmerican Journal of Mathemaics 130 (2008), 385-412
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/161492
dc.subjectGeometric Topology
dc.subjectDifferential Geometry
dc.subjectDynamical Systems
dc.subject57M05; 30F60; 20H10; 37F30
dc.titleEnd Invariants for $\SL(2,C)$ characters of the one-holed torus
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