Thompson's Group F
Abstract
We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an orderpreserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for braid groups, we present a classifying space for F that is the ``configuration space'' of finitely many points on a line, with the points allowed to split and merge in pairs. Strand diagrams are related to a description of F as a groupoid, which we use to derive presentations for F, T, V, and the braided Thompson group BV. In addition to the new results, we include a thorough exposition of the basic theory of the group F. Highlights include a simplified proof that the commutator subgroup of F is simple, a discussion of open problems (with a focus on amenability), and a simplified derivation of the standard presentation and normal forms for F using forest diagrams.
 Publication:

arXiv eprints
 Pub Date:
 August 2007
 arXiv:
 arXiv:0708.3609
 Bibcode:
 2007arXiv0708.3609B
 Keywords:

 Mathematics  Group Theory;
 20F65
 EPrint:
 Ph.D. Thesis (Cornell University). Includes arXiv:math/0305412 and arXiv:math/0301141, as well as much additional material