Geodesics and patterns in some interesting groups
Abstract: We start with the Contiki tour of Geometric and Automatic group theories (that is, just the highlights in 10 minutes): For every group and finite generating set we can draw a "Cayley graph". A word in the generators of the group corresponds to a path in the graph. A geodesic word is a shortest path between two points in the graph. We define an "automatic group" which finds connections between group theory and theoretical computer science.
We then take a group from an intersting class of groups, examine in detail its geometry and make some surprising statements about its geodesic automatic structure.