Conjugacy classes in M\"obius groups
Krishnendu Gongopadhyay

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 -dimensional (real) hyperbolic space. Let denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of is denoted by . Let be its identity component which consists of all orientation-preserving elements in . The conjugacy classification of isometries in depends on the conjugacy of and in . For an element in , and are conjugate in , but they may not be conjugate in . In the literature, is called real if is conjugate in to . In this paper we classify real elements in . Let be an element in . Corresponding to there is an associated element in . If the complex conjugate eigenvalues of are given by , $0 <…
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