Fundamental domains for properly discontinuous affine groups

TL;DR

This paper constructs fundamental regions for certain free, discrete, and properly discontinuous affine groups acting on high-dimensional affine spaces, specifically those preserving a quadratic form with signature (d+1, d).

Contribution

It introduces a method to explicitly construct fundamental domains for affine groups preserving quadratic forms of signature (d+1, d), applicable for any odd positive integer d.

Findings

Constructed fundamental regions for affine groups in 2d+1 dimensions.

Applicable to groups preserving quadratic forms of signature (d+1, d).

Provides a framework for understanding the geometry of these affine actions.

Abstract

We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.