# Imaginary cones and limit roots of infinite Coxeter groups

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## Abstract

Let (*W*, *S*) be an infinite Coxeter system. To each geometric representation of *W* is associated a root system. While a root system lives in the positive side of the isotropic cone of its associated bilinear form, an imaginary cone lives in the negative side of the isotropic cone. Precisely on the isotropic cone, between root systems and imaginary cones, lives the set *E* of limit points of the directions of roots. In this article we study the close relations of the imaginary cone with the set *E*, which leads to new fundamental results about the structure of geometric representations of infinite Coxeter groups. In particular, we show that the *W*-action on *E* is minimal and faithful, and that *E* and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of *W*, i.e., root systems for Coxeter groups without braid relations (the free object for Coxeter groups). Finally, we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory.

## Keywords

Coxeter group Root system Roots Limit point Accumulation set Elementary roots Small roots## Mathematics Subject Classification

Primary 20F55 Secondary 17B22 37B05## Notes

### Acknowledgments

The authors wish to thank Jean-Philippe Labbé who made the first version of the Sage and TikZ functions used to compute and draw the normalized roots. The third author gave, in France in November 2012, several seminar talks about a preliminary version of these results; he is grateful to the organizers of these seminars and to the participants for many useful comments. In particular, he would like to thank Vincent Pilaud for valuable discussions. The authors also wish to thank an anonymous referee for valuable suggestions that improved the present manuscript, especially 7.4, and that led to the Appendix.

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