Redundancy of triangle groups in spherical CR representations

TL;DR

This paper investigates boundary unipotent CR representations of non-compact three-manifold groups, identifying conditions under which they are discrete, related to (3,3,n) triangle groups, and highlighting redundancy among these representations.

Contribution

It verifies the existence of (3,3,n) triangle group representations with boundary unipotent holonomy and demonstrates redundancy, showing only one is uniformizable for fixed n.

Findings

Representations with fractal limit sets are discrete.

Many (3,3,n) representations are conjugate and redundant.

Only one representation per n is uniformizable.

Abstract

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the (3,3,n) representations encountered are conjugate and only one among them is uniformizable.