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When Abelian Groups Split, Rachel M. Thomas, Robert C. Rhoades

Mathematical Sciences Technical Reports (MSTR)

Let S be a hyperbolic surface tiled by kaleidoscopic triangles. Let R_e denote the set of fixed points by the reflection in an edge, e, of a triangle. We say that R_e is separating if S-R_e 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 …