S. Allen Broughton
Let S be a hyperbolic surface tiled by kaleidoscopic triangles. Let Re denote the set of fixed points by the reflection in an edge, e, of a triangle. We say that Re is separating if S-Re has two components. Once we have a tiling, we can define a group of orientation preserving transformations, G. We develop a method for determining when a reflection is separating using the group algebra of G. Using this method we give necessary and sufficient conditions for a mirror to be separating when G is abelian. We also conjecture, that when G is simple there are no separating mirrors.
Thomas, Rachel M. and Rhoades, Robert C., "When Abelian Groups Split" (2003). Mathematical Sciences Technical Reports (MSTR). 46.