A Note on Discrete Groups

TL;DR

This paper characterizes groups with a DF domain as those with a Double Dirichlet domain, describing their side-paring transformations and eigenvectors, with additional results on Kleinian and Fuchsian groups.

Contribution

It establishes an equivalence between DF domains and Double Dirichlet domains in groups, and details their geometric and algebraic properties.

Findings

Groups with DF domains are exactly those with Double Dirichlet domains.

Such groups not fixing infinity share a common eigenvector.

Provides new insights into Kleinian and Fuchsian group structures.

Abstract

We prove that a group has a DF domain if and only if has a Double Dirichlet domain. We give a good describtion of the side-paring transformation of such groups. In particular, those not fixing infinity have a common eigenvector. We also give other results on Kleinian and Fuchsian groups.