Möbius群中的共轭类
Conjugacy classes in Möbius groups
摘要：
设$\H^{n+1}$表示$n + 1$维（实）双曲空间.设$\s^{n}$表示双曲空间的共形边界。$\s^n$的共形仿射群记为$M (n)$。令$M_o (n)$是它的身份组成部分，它由$M (n)$中的所有方向保持元素组成。$M_o (n)$中等距体的共轭分类取决于$M_o (n)$中$T$和$T^{-1}$的共轭性。对于$M (n)$中的元素$T$，$T$和$T^{-1}$在$M (n)$中共轭，但它们在$M_o (n)$中可能不共轭。在文献中，如果$T$在$M_o (n)$中与$T^{-1}$共轭，则称$T$为实数。本文对$M_o (n)$中的实元素进行了分类。让$T$成为$M_o(n)$中的一个元素。对应于$T$，在$SO(n+1)$中存在缔合元素$T_o$。如果$T_o$的复共轭特征值由$\{e^{iθ_j}, e^{-iθ_j}\}$，$0 < θ_j \leq π$，$j=1,...,k$给出，则$\{θ_1,...,θ_k\}$称为$T$的\emph{rotation angles}.如果$T$的旋转角度彼此不同，则$T$称为\emph{regular}元素。在对$M_o (n)$中的实元素进行分类之后，我们对$M_o (n)$中正则元素的共轭类进行了参数化。在参数化中，当$T$与$T^{-1}$不共轭时，我们扩大了群，并考虑了$T$在$M (n)$中的共轭类。我们证明了每个这样的共轭类可以诱导纤维化结构。
英文摘要：
Let $\H^{n+1}$ denote the $n + 1$-dimensional (real) hyperbolic space. Let $\s^{n}$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of $\s^n$ is denoted by $M (n)$. Let $M_o (n)$ be its identity component which consists of all orientation-preserving elements in $M (n)$. The conjugacy classification of isometries in $M_o (n)$ depends on the conjugacy of $T$ and $T^{-1}$ in $M_o (n)$. For an element $T$ in $M (n)$, $T$ and $T^{-1}$ are conjugate in $M (n)$, but they may not be conjugate in $M_o (n)$. In the literature, $T$ is called real if $T$ is conjugate in $M_o (n)$ to $T^{-1}$. In this paper we classify real elements in $M_o (n)$. Let $T$ be an element in $M_o(n)$. Corresponding to $T$ there is an associated element $T_o$ in $SO(n+1)$. If the complex conjugate eigenvalues of $T_o$ are given by $\{e^{iθ_j}, e^{-iθ_j}\}$, $0 < θ_j \leq π$, $j=1,...,k$, then $\{θ_1,...,θ_k\}$ are called the \emph{rotation angles} of $T$. If the rotation angles of $T$ are distinct from each-other, then $T$ is called a \emph{regular} element. After classifying the real elements in $M_o (n)$ we have parametrized the conjugacy classes of regular elements in $M_o (n)$. In the parametrization, when $T$ is not conjugate to $T^{-1}$, we have enlarged the group and have considered the conjugacy class of $T$ in $M (n)$. We prove that each such conjugacy class can be induced with a fibration structure.
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