Conjugacy classes in M\"obius groups

Let \H^{n+1} denote the $n+1$-dimensional (real) hyperbolic space. Let ${\text{s}}^n$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of ${\text{s}}^n$ is denoted by $M(n)$. Let $M_o(n)$ be its identity component which consists of all orientation-preserving elements in $M (n)$\mathrm{.}Theconjugacyclassificationofisometriesin$M_o (n)$ 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 $M_o (n)$\mathrm{.}Intheliterature,$T$iscalledrealif$T$isconjugatein$M_o (n)$to$T^{-1}$\mathrm{.}Inthispaperweclassifyrealelementsin$M_o (n)$. 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 $\{e^{i{\theta }_j},e^{-i{\theta }_j}\}$, $0 < \theta_j \leq \pi$,$j=1,…,k$,then${\theta_1,…,\theta_k}$ 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.