  Given any Coxeter group, we define rigid reflections and rigid roots using
non-self-intersecting curves on a Riemann surface with labeled curves. When the
Coxeter group arises from an acyclic quiver, they are related to the rigid
representations of the quiver. For a family of rank 3 Coxeter groups, we show
that there is a surjective map from the set of reduced positive roots of a rank
2 Kac--Moody algebra onto the set of rigid reflections. We conjecture that this
map is bijective.
