Generalized Hyperbolic Spaces Associated with Arbitrary Quadratic Forms

TL;DR

This paper extends the concept of hyperbolic spaces associated with quadratic forms by defining generalized spaces over arbitrary fields, enabling Vahlen groups to act transitively via Möbius transformations.

Contribution

It introduces a framework for hyperbolic spaces linked to any quadratic form over fields of characteristic not 2, broadening the scope of Vahlen groups and their geometric actions.

Findings

Defined generalized hyperbolic spaces for arbitrary quadratic forms

Extended Vahlen groups to operate over any field of characteristic not 2

Established boundary components for Möbius transformations in these spaces

Abstract

The Vahlen group gives a way for presenting the hyperbolic space of every dimension of a group acting via M\"{o}bius transformations. As Vahlen groups and paravector Vahlen groups are now defined over any field of characteristic different from 2, we establish analogous spaces on which they operate transitively as M\"{o}bius transformations, by defining appropriate boundary components that must be added in case the denominator in the M\"{o}bius transformation formula vanishes.