Conjugacy Classes in Möbius groups

Abstract

Let ℍn+1 denote the n + 1-dimensional (real) hyperbolic space. Let Sn denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of Sn 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 {eiθj, e-iθj}, 0 < θj ≤ π, j = 1,. . ., k, 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