Date of this Version
UCARE Research Fair, Spring 2016, University of Nebraska-Lincoln
Take a square and flip it over the vertical axis, rotate it 90 degrees counterclockwise and then flip it again over the vertical axis. This sequence is the same as a 90 degree clockwise rotation but takes more steps to demonstrate the same symmetry. In general, the question of when a sequence of symmetries has minimal length is hard to answer and is dependent on the chosen generating set (in our toy example, rotation by 90 degrees and reflection). By realizing sequences of symmetries as paths in a group's Cayley graph, the problem becomes one about the set of shortest paths in a metric space and admits many geometric methods of approach. The language theoretic properties of the set of geodesics on the Cayley graph are related to many classical problems in geometric group theory, including automaticity, rationality of the group's growth series and the solvability of the word problem. This poster will concern results on geodesics and geodesics up to conjugacy in Coxeter groups, generalized groups of reflections.