On the limit set of root systems of Coxeter groups acting on Lorentzian spaces

Abstract

The notion of limit roots of a Coxeter group W was recently introduced: they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a Lorentzian space, i.e., W admits a faithful representation as a discrete reflection group of isometries of a hyperbolic space, the accumulation set of any of its orbits is then classically called the limit set of W. In this article, we show that the set of limit roots of a Coxeter group W acting on a Lorentzian space is equal to the limit set of W seen as a discrete reflection group of hyperbolic isometries.

Keywords:

• Root system
• Coxeter group
• limit roots
• limit set of discrete group
• Lorentzian space
• discrete reflection group
• Apollonian gasket
• hyperbolic Coxeter group
• hyperbolic isometries
• Kleinian groups

MATHEMATICS SUBJECT CLASSIFICATION:

• 20F55
• 51F15
• 17B22
• 30F40

Acknowledgments

The authors wish to thank Jean-Philippe Labbé who made the first version of the Sage [28] and TikZ functions used to compute and draw the normalized roots. The second author wishes to thank Pierre de la Harpe for his invitation to come to Geneva in June 2013 and for his comments on this article. The third author is grateful to Pierre Py for fruitful discussions in Strasbourg in November 2012. We also acknowledge the participation of Nadia Lafrenière and Jonathan Durand Burcombe to a LaCIM undergrad summer research award on this theme during the summer 2012. The authors wish to warmly thank the anonymous referee for his/her helpful comments that improved the quality of this article.

Notes

1 This vocabulary is inspired from the theory of relativity, where n = 3.

2 These transformations are called B-isometries in [9, 17], since in these articles B does not necessarily have signature $(n,1).$

3 Observe that if B was positive definite, this equation would be the usual formula for a Euclidean reflection.

4 This model is also sometimes called the Beltrami-Klein model in the literature.

5 From Corollary 2.3.

6 Also called fundamental convex polyhedron in the literature, see [26, 30].

7 See Remark 3.2.

8 This means that the Coxeter graph is connected and that the Coxeter group is neither finite nor affine, see for instance [9] for more details.

9 It contains all reflections with respect to hyperplanes that are perpendicular to the boundary.

10 It contains all reflections with respect to hyperplanes perpendicular to ${\mathbb{R}}^{n-1}\times \left\{0\right\}.$

Additional information

Funding

During this work the first author was supported by a NSERC grant and the third author was supported by a postdoctoral fellowship from LaCIM. This collaboration was also made possible with the support of the UMI CNRS-CRM.