Visualizing Geometric Structures on Topological Surfaces

We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the mapping class group acts on geometric structures. We explore the geometry of closed, compact, and orientable two-dimensional manifolds through direct visualization and computation. We prove that the mapping class group of a torus is isomorphic to SL2Z via direct matrix multiplication on the generating elements of the fundamental group. While the fundamental group of the torus has only one possible presentation, up to homeomorphism; the case for the genus 2 surface is more complicated. We prove that an octagon representing a genus 2 surface can have its edges identified in different combinations to produce exactly four different possible presentations of fundamental groups. We explore surgeries on one of those types and show that surgeries that preserve that type are equivalent to Dehn twists on the surface, which are generators of the mapping class grou