## Cayley graph

Our photo this week is by Christian Perfect, a PhD student at the University of Newcastle. He was drawing a Cayley graph, which is a picture encoding the structure of a group. This particular picture is a drawing of the structure of F_{2}, the free group on two generators. This group is generated by the elements {x,y,x^{-1}, y^{-1}} with x and x^{-1} being right and left arrows, and y and y^{-1} being up and down arrows respectively. Each element of the group is then a vertex of the graph. For example, to find the vertex corresponding to xyx^{2}y^{-1} you start in the middle and go right, up, right, right, down.

This picture can also be interpreted (if you are a topologist!) as the universal cover of the ‘8’, or infinity, symbol.

Christian actually took a series of photos as he was drawing the graph. I’ve put these together into an animated gif for your viewing pleasure (but beware – it’s 30Mb!).

Andrew RanickiThere is a thickened Cayley graph on the blackboard in the photo of Des Sheiham in http://www.maths.ed.ac.uk/~aar/des.htm

August 8, 2011 at 9:23 pm