Mathematics – Group Theory
Scientific paper
2007-08-26
Mathematics
Group Theory
Ph.D. Thesis (Cornell University). Includes arXiv:math/0305412 and arXiv:math/0301141, as well as much additional material
Scientific paper
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 order-preserving 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.
No associations
LandOfFree
Thompson's Group F does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Thompson's Group F, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Thompson's Group F will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-600059