Jim Belk

Research

I am interested primarily in combinatorial and geometric group theoryWikipedia. My research so far has focused on the following topics:

Thompson's group FWikipedia (the structure group of associativity), and various related groups.
Amenability of groupsWikipedia, and its relation to the geometric structure of Cayley graphsWikipedia.
Groups of automata and self-similar groups, e.g. Grigorchuk's group of intermediate growthWikipedia.

I am just now learning about this last topic, though hopefully I'll have something original to say about it soon.

Colleagues

I received my Ph.D from Cornell University in August of 2004. My thesis adivsor was was Ken Brown, and I also worked closely at Cornell with Kai-Uwe Bux and Francesco Matucci.

I'm here at Texas A&M for three years to work with Rostislav Grigorchuk, Volodymyr Nekrashevych, and Zoran Sunik. My position at Texas A&M is partially funded by an NSF postdoctoral research fellowship.

Papers

I have published two research papers:

James M. Belk and Kenneth S. Brown. Forest Diagrams for Elements of Thompson's Group FarXiv preprint. Internat. J. Algebra Comput. 15 (2005), 815-850.
James Belk and Kai-Uwe Bux. Thompson's Group F is Maximally NonconvexarXiv preprint. Geometric Methods in Group Theory, Contemp. Math. 372, Amer. Math. Soc. (2005), 131-146.

I am currently working on a new paper with Francesco Matucci concerning the conjugacy problem in Thompson's groups F, T, and V.

Ph.D. Thesis

My Ph.D. thesis includes material from my two published papers, a nice exposition of Thompson's group, and some additional research that has not yet been published elsewhere. You can download my thesis in in postscript or PDF format.

I am very proud of the fact that my thesis has 157 pictures on 132 pages.

If you've read my two papers, you may be interested in the following parts of my thesis:

The proof that [F, F] is simple (pg. 16) is somewhat different from previously published proofs of this fact.
The material on one-way forest diagrams (pgs. 31-48) has not been published elsewhere. In particular, the material on pgs. 42-48 is related to some research by Zoran Sunik on a relationship between Thompson's group F and Tamari lattices.
The proof that the isoperimetric constant of F is at most 1/2 (pgs. 86-92) has not been published elsewhere.
None of the material on strand diagrams (pgs. 115-132) has yet been published, though some of this should be covered in my upcoming paper with Francesco.

Talks

Here are the powerpoint slides for my forest diagrams talk. Among other things, the slides show several animations of forest diagrams being constructed, which may prove helpful to anyone interested in my papers. I have given some version of this talk five times:

As a plenary talk (joint with Ken Brown) during the International Conference on Group Theory in Gaeta, Italy (June 2003).
At a special session on geometric group theory, during a fall 2003 AMS meeting at Binghamton Unviversity (20 minute talk).
During my thesis defense at Cornell in August of 2004.
At the Binghamton Unviversity Topology/Geometry seminar in the fall of 2004.
In the Groups and Dynamics seminar here at Texas A&M (fall of 2005).

Last year I gave a short talk to the Texas A&M math department describing my area of research. You can download the slides here.

Last spring I gave a second talk in the Groups and Dynamics seminar entitled Strand Diagrams, Configuration Spaces, and Associahedra, but it did not use powerpoint, and there is not currently any downloadable version.

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