Poincare Extension of Moebius Transformations

TL;DR

This paper explores the algebraic foundations of the Poincaré extension of Möbius transformations, focusing on their sphere-preserving properties and describing their structure using one-parameter subgroups in a simplified one-dimensional setting.

Contribution

It provides a comprehensive algebraic analysis of the Poincaré extension for Möbius transformations, detailing all available options and focusing on the real line case.

Findings

Algebraic description of Poincaré extension using quadratic forms

Characterization of one-parameter subgroups of Möbius transformations

Simplification to the real line case for conceptual clarity

Abstract

Given sphere preserving (M\"obius) transformations in $n$-dimensional Euclidean space one can use the Poincar\'e extension to obtain sphere preserving transformations in a half space of $n+1$ dimensions. The Poincar\'e extension is usually provided either by an explicit formula or by some geometric construction. We investigate its algebraic background and describe all available options. The solution is given in terms of one-parameter subgroups of M\"obius transformations acting on triples of quadratic forms. To focus on the concepts, this paper deals with the M\"obius transformations of the real line only.