Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 28th of January 2009, 3:00-3:50
Groups and Dynamics Seminar
Groups related to Thompson's group F
Dmytro Savhuk of Texas
A&M
University
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 action on the Cantor set 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.