  Results are obtained concerning the roots of asymmetric geometric
representations of Coxeter groups. These representations were independently
introduced by Vinberg and Eriksson, and generalize the standard geometric
representation of a Coxeter group in such a way as to include all Kac--Moody
Weyl groups. In particular, a characterization of when a non-trivial multiple
of a root may also be a root is given in the general context. Characterizations
of when the number of such multiples of a root is finite and when the number of
positive roots sent to negative roots by a group element is finite are also
given. These characterizations are stated in terms of combinatorial conditions
on a graph closely related to the Coxeter graph for the group. Other finiteness
results for the symmetric case which are connected to the Tits cone and to a
natural partial order on positive roots are extended to this asymmetric
setting.
