Conjugacy classes in Möbius groups

Abstract

Let ${\mathbb H^{n+1}}$ denote the n + 1-dimensional (real) hyperbolic space. Let ${\mathbb {S}^{n}}$ denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of ${\mathbb {S}^{n}}$ 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 ${\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}$ , then {θ₁, . . . , θ _k } 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 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.