Conjugacy classes in Möbius groups

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb H^{n+1}}$$\end{document} denote the n + 1-dimensional (real) hyperbolic space. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}^{n}}$$\end{document} denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}^{n}}$$\end{document} is denoted by M(n). Let Mo(n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in Mo(n) depends on the conjugacy of T and T−1 in Mo(n). For an element T in M(n), T and T−1 are conjugate in M(n), but they may not be conjugate in Mo(n). In the literature, T is called real if T is conjugate in Mo(n) to T−1. In this paper we classify real elements in Mo(n). Let T be an element in Mo(n). Corresponding to T there is an associated element To in SO(n + 1). If the complex conjugate eigenvalues of To are given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}$$\end{document} , then {θ1, . . . , θk} are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in Mo(n) we have parametrized the conjugacy classes of regular elements in Mo(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.