On the Limit Set of Root Systems of Coxeter Groups acting on Lorentzian spaces

TL;DR

This paper establishes that the limit roots of Coxeter groups acting on Lorentzian spaces are equivalent to the classical limit set of the group as a hyperbolic isometry group, bridging algebraic and geometric perspectives.

Contribution

It proves the equality between the algebraic limit roots and the geometric limit set for Coxeter groups in Lorentzian spaces, unifying two approaches.

Findings

Limit roots coincide with the hyperbolic limit set.

Provides a self-contained proof accessible to both reflection group and hyperbolic geometry communities.

Bridges algebraic and geometric frameworks for Coxeter groups.

Abstract

The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): 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 W admits a faithful representation as a discrete reflection group of isometries on 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 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. We aim for this article to be as self-contained as possible in order to be accessible to the community familiar with reflection groups and root systems and to the community familiar with discrete subgroups of isometries in hyperbolic geometry.