Free groups generated by two parabolic maps

TL;DR

This paper investigates conditions under which a group generated by two parabolic elements in SU(2,1) is discrete and free, and explores properties of finite real circles in the Heisenberg group.

Contribution

It provides new criteria for discreteness and freeness of groups generated by two parabolic maps in SU(2,1), and analyzes geometric properties of real circles in the Heisenberg group.

Findings

Conditions guaranteeing group discreteness and freeness.

Results on the diameter of finite real circles in the Heisenberg group.

Abstract

In this paper we consider a group generated by two unipotent parabolic elements of ${\rm SU}(2,1)$ with distinct fixed points. We give several conditions that guarantee the group is discrete and free. We also give a result on the diameter of a finite ${\mathbb R}$-circle in the Heisenberg group.