Redundancy of Triangle Groups in Spherical CR Representations

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.

Keywords:

• Boundary unipotent representations
• Complex hyperbolic triangle groups
• Knot link complements
• Limit sets
• Spherical CR representations

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Acknowledgments

This paper is part of the author’s thesis, in progress under the supervision of Elisha Falbel. The author enjoyed many and very fruitful conversations. First of all, of course, I am very thankful to Elisha Falbel. Since the early stages in making the experimental tools, Fabrice Rouillier and Antonin Guilloux have been of precious help for the improvement of my code. Across countries, Mathias Görner has been essential to me in order to correctly use the tools provided by SnapPy. I would like to thank Pierre Will for all his comments and the many hours of discussions we had. I have been pleased to exchange with Miguel Acosta about his theorem and about experimental aspects. Finally, I would like to thank the reviewers who gave me many comments significantly improving the text.

Notes

1 We also observed that every representation from the census that gave a fractal had each boundary component (a copy of ${\mathbf{Z}}^2$) sent to a rank one parabolic subgroup.