Rigid Reflections of Rank 3 Coxeter Groups and Reduced Roots of Rank 2 Kac-Moody Algebras

In a recent paper by K.-H. Lee and K. Lee, rigid reflections are defined for any Coxeter group via non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, the rigid reflections are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, it was conjectured in the same paper that there is a natural bijection from the set of reduced positive roots of a symmetric rank 2 Kac-Moody algebra onto the set of rigid reflections of the corresponding rank 3 Coxeter group. In this paper, we prove the conjecture.