Abstract: In the past I have given several lectures on kaleidoscopic tilings by triangles and quadrilaterals on surfaces, and asserted in these talks that the tiling group completely determined the combinatorial and topological structure of a tiling. However, I have never really talked about the influence of the group theory! In this series of two talks I will give two examples of determining combinatorial and topological structure, by group computations. Each talk will focus on a problem I intend to give to REU students this summer. Thus, there will be no general theorems just problems statements with suggestions of attack, the talks will focus on developing the background to get to the problem statements. The first talk will include the necessary review of tilings and hyperbolic geometry. You don't need to know much about group theory or hyperbolic geometry.
First talk: Constructing a fundamental domain for kaleidoscopically tiled surfaces. We are all familiar with the process of creating a torus by identifying opposite sides of a euclidean rectangle. For higher genus surfaces of genus s > 1, a surface may be constructed by identifying sides of a hyperbolic 4s-gon. For a kaleidoscopically tiled surface can this be done so that the polygon is a "nice" collection of tiles? The group theory computation will be focus on relating the infinite tiling group on the hyperbolic plane to the finite tiling group on the surface.
Second Talk: When are kaleidoscopic tilings separating? Every edge of a kaleidoscopic tiling generates a reflection of the surface to itself fixing the edge. In the case of a sphere the fixed point set (or mirror) of the reflection is a great circle which separates the sphere into two pieces. This is very misleading example, since for higher genus the mirror very rarely separates the surface. The question is: is there a fast way to determine this splitting property from the properties of the tiling group? The talk will present a method of attack using the group algebra of the talk. Again, no previous knowledge of group theory is assumed.