arXiv math/05116212005-11-25math.GTmath.DGmath.DS

单孔环面的$\SL(2,C)$特征的端点不变量

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

Ser Peow Tan, Yan Loi Wong, Ying Zhang

备注：: 24 pages, 6 figures
发表信息：: American Journal of Mathemaics 130 (2008), 385-412

摘要：

定义并研究了单孔环面$T$的$\SL(2,C)$特征$ρ$的端不变量集${\mathcal E}(ρ)$.证明了集合${\mathcal E}(ρ)$是$T$的全射影层空间$\mathscr{PL}$当且仅当（i）$ρ$对应于二面角表示，或（ii）$ρ$是实数且对应于SU（2）表示;否则，${\mathcal E}(ρ)$是闭的且在$\mathscr{PL}$中有空内部.对于实字符$ρ$，我们给出了${\mathcal E}(ρ)$的完全分类，并证明了${\mathcal E}(ρ)$有0、1或无穷多个元素，在最后一种情况下，${\mathcal E}(ρ)$要么是$\mathscr{PL}$的Cantor子集，要么是$\mathscr{PL}$本身.我们也给了一个类似的分类“虚”字符的换向器的痕迹是小于2。最后，我们证明了对于离散特征标（不对应于二面角或SU（2）表示），如果${\mathcal E}(ρ)$包含至少三个元素，则它是$\mathscr{PL}$的Cantor子集。

英文摘要：

We 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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