#### Document Type

Article

#### Publication Date

8-1-2003

#### First Advisor

S. Allen Broughton

#### Abstract

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*has two components. Once we have a tiling, we can define a group of orientation preserving transformations,

_{e}*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.

#### Recommended Citation

Thomas, Rachel M. and Rhoades, Robert C., "When Abelian Groups Split" (2003). *Mathematical Sciences Technical Reports (MSTR)*. 46.

http://scholar.rose-hulman.edu/math_mstr/46

## Comments

MSTR 03-01