Wallpapering the Hyperbolic Plane: Discrete Subgroups Of the Euclidean and Hyperbolic Planes

Abstract

The question driving this thesis is, \lq\lq{}What are the hyperbolic analogues of the Euclidean wallpaper groups?\rq\rq{}We first investigate symmetry groups of the Euclidean plane, the frieze and wallpaper groups, understanding both in terms of generators and relations. All wallpaper groups have presentations as an extension of a point group by $\Z^2$.Some also have presentations in terms of reflections in the sides of a triangle. Since triangle groups also exist in the hyperbolic plane, we discuss the Euclidean triangle groups.%This is the gist of the chapters set in the Euclidean plane. %\textbf{Isometries in $\EE$}.We then examine the isometries of the hyperbolic plane, finding that Isom$\f\HH\g$ differs structurally from Isom$\f\EE\g$.In particular, translations are not closed in the hyperbolic plane.Thus, though the frieze groups are structurally the same in the hyperbolic as in the Euclidean context, Isom$\f\HH\g$ has no translation subgroup, and so the Euclidean wallpaper groups do not transfer.%Because of these structural differences, the frieze groups map directly between the Euclidean and hyperbolic planes, but the Euclidean wallpaper groups do not. %There are no direct \lq\lq{}hyperbolic versions\rq\rq{} of Euclidean wallpaper groups, and yet However, there do exist tilings of the hyperbolic plane.We close with a discussion of the Fuchsian groups, and some hyperbolic triangle groups.%the orientation-preserving isometry groups of the hyperbolic plane which have a fundamental region with which they tile the hyperbolic plane. The original results of this thesis analyze the composition of certain translations and of certain parallel displacements in Isom$\f\HH\g$.