DATE: | February 5, 2009 |
TIME: | 4:30 - 5:30 PM |
PLACE: | LN 2205 |
SPEAKER: | Dmytro Savchuk, Texas A&M University |
TITLE: | Graphs related to Thompson's group F |
Perhaps, the most intriguing open question about Thompson's group F is whether or not it is amenable. We approach this question from two different points of view.
On the one hand we explicitly construct the Schreier graphs of Thompson's group F with respect to the stabilizers of each irrational or dyadic rational point of the interval [0,1] and the standard generating set {x_0, x_1}, and show that these graphs are amenable.
On the other hand we describe the structure of an induced subgraph of the Cayley graph of F with respect to the generating set {x_0,x_1}, containing all vertices of the form x_nw for w in the monoid generated by x_0 and x_1 and n>=0. We show that this graph is non-amenable.
Unfortunately, none of the above approaches gives the answer to the ultimate question about the amenability of F, but both shed some light on the structure of the group itself.