Benoist-Hulin groups

TL;DR

This paper explores Benoist-Hulin groups, a special class of subgroups of PSL(2,C), demonstrating that uniform lattices and parabolic subgroups belong to this class, thus expanding the understanding of their structure.

Contribution

The paper develops the theory of Benoist-Hulin groups and proves that uniform lattices and parabolic subgroups are examples of such groups.

Findings

Full group PSL(2,C) is a Benoist-Hulin group.

Uniform lattices are Benoist-Hulin groups.

Parabolic subgroups are Benoist-Hulin groups.

Abstract

A Benoist-Hulin group is, by definition, a subgroup $\Gamma$ of ${\rm PSL}_2(\mathbb{C})$ such that any $\Gamma$-invariant closed set consisting of Jordan curves in the space of closed subsets of the Riemann sphere that are not singletons is composed of $K$-quasicircles for some $K \ge 1$. Y.Benoist and D.Hulin showed that the full group ${\rm PSL}_2(\mathbb{C})$ is a Benoist-Hulin group. In this paper, we develop the theory of Benoist-Hulin groups and show that both uniform lattices and parabolic subgroups are Benoist-Hulin groups.