  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 isotropy 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 (see arXiv:1112.5415). In this article we study the
close relations of the imaginary cone (see arXiv:1210.5206) 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.
