  Brink and Howlett have introduced a partial ordering, called dominance, on
the positive roots in the Tits realization of Coxeter groups (Math. Ann. 296
(1993), 179--190). Recently a concept called $\infty$-height is introduced to
each reflection in an arbitrary Coxeter group $W$ (Edgar, Dominance and
regularity in Coxeter groups, PhD thesis, 2009). It is known (Dyer,
unpublished) that for all $W$ of finite rank, and for each non-negative $n$,
the set of reflections of $\infty$-height equal to $n$ is finite. However, it
is not clear that the concepts of $\infty$-height and dominance are related.
Here we show that the $\infty$-height of an arbitrary reflection is equal to
the number of positive roots strictly dominated by the positive root
corresponding to that reflection. We also give applications of dominance to the
study of imaginary cones of Coxeter groups.
