Conjugacy classes in M\"obius groups

TL;DR

This paper classifies real conjugacy classes of isometries in the orientation-preserving conformal group of hyperbolic space, providing a parametrization for regular elements and revealing a fibration structure for certain conjugacy classes.

Contribution

It offers a detailed classification of real elements in the Möbius group of hyperbolic space and parametrizes conjugacy classes of regular elements, including a new fibration perspective.

Findings

Classification of real elements in $M_o(n)$.

Parametrization of conjugacy classes of regular elements.

Identification of a fibration structure in conjugacy classes.

Abstract

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 $\{e^{i\theta_j}, e^{-i\theta_j}\}$, $0 < \theta_j \leq \pi$, $j=1,...,k$, 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.