1. I am interested in geometric group theory and low dimensional topology. Specific topics include hyperbolic groups (and their subgroups), free-by-free/free-by-cyclic groups, Artin groups, CAT(0) groups, Morse theory, (high dimensional) filling invariants of groups, finiteness properties of groups, automatic and biautomatic groups, cubical complexes...

  2. Here is a list of my papers and preprints.

  3. I have one graduate student who is currently thinking about topics in geometric group theory;
    Sang Rae Lee.

  4. In Spring 2008 my first student, Eduardo Martinez-Pedroza, completed his PhD thesis on combination theorems for quasi-convex subgroups of relatively hyperbolic groups. He is currently a post-doctoral associate at McMaster University.

  5. In Fall 2008 my student Antara Mukherjee completed her PhD thesis on second order Dehn functions of Nil and Sol lattices and Varopolous transport. She is currently a lecturer at the Citadel in Charlestown, South Carolina.

  6. In Summer 2009 my student TaraLee Mecham completed her PhD thesis on largeness of graphs of abelian groups. She is currently an assistant professor of mathematics at the College of Mount St Joseph in Cincinnati, Ohio. She is also a Project NExT fellow.

  7. Current and past geometric group theory post-doctoral associates at the University of Oklahoma.

  8. You can see the current schedules of the OU Geometry and Topology Seminar and of the Student Topology and Geometry Seminar.

  9. If you are interested in learning about geometric group theory, here are some good online resources.

  10. By now there is an extensive list of survey articles and books on geometric group theory and related topics. Here is a (necessarily incomplete) selection. This is under construction.

      Background.

    • It is important to have a solid foundation in algebraic topology, particularly fundamental groups and covering spaces.

      Some Historical Results/Development.

    • Max Dehn.
    • John Stallings was one of the first people to explore the deep connection between 3-manifold topology and the theory of finitely presented groups. Now we have results on accessibility, algebraic annulus and torus theorems, and the JSJ decomosition theory. Stallings' paper A topological proof of Grushko's theorem on free products in Math. Z. 90 1965 1-8 is a great example of this connection. So also is On fibering certain $3$-manifolds , 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95--100 Prentice-Hall, Englewood Cliffs, N.J.
    • Stallings was also one of the pioneers of the idea that algebraic information can be encoded in the coarse geometric/topological structure of the group. His On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 1968 312--334 is an excellent example. Stallings' Topology of finite graphs which appeared in Invent. Math. 71 (1983), no. 3, 551--565 is a wonderful example of elegance, simplicity and depth.
    • William P Thurston. The Geometry and Topology of Three-Manifolds by W P Thurston. Electronic (scanned) version of Thurston's 1980 Lecture Notes. Excellent stuff.
    • Mikhail Gromov. Infinite groups as geometric objects ICM address in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), 385--392, PWN, Warsaw, 1984. This lays out the foundations of geometric group theory.
    • Gromov's Groups of polynomial growth and expanding maps, Publ. Math. IHÉS 53 (1981), 53-73.
    • M Gromov's seminal paper Hyperbolic Groups in "Essays in Group Theory" (G. M. Gersten, ed.), MSRI Publ. 8, 1987, pp. 75-263.
    • Another seminal contribution of M Gromov to the field: Asymptotic invariants of infinite groups in Geometric Group Theory, Vol 2 edited by G Niblo and M Roller.

      Survey Articles.

    • There is a nice article by Scott and Wall called Topological Methods in Group Theory in Homological Group Theory London Mathematical Society Lecture Note Series (No. 36).
    • Geometric Group Theory survey article by J W Cannon in the Handbook of Geometric Topology edited by R J Daverman and R B Sher. Indeed, there are many great survey articles in this collection.
    • The geometries of 3-manifolds. by P Scott. Bull. London Math. Soc. 15 (1983), no. 5, 401--487.
    • A course on geometric group theory by B H Bowditch.
    • Martin Bridson has some good survey articles. One on Non-Positive Curvature in Group Theory in Groups St. Andrews 1997 in Bath, I, 124--175, London Math. Soc. Lecture Note Ser., 260, Cambridge Univ. Press, Cambridge, 1999, and one on The Geometry of the Word Problem in Invitations to Geometry and Topology edited by M R Bridson and S M Salamon.
    • Hyperbolic Geometry survey by Cannon-Floyd-Kenyon-Parry in Flavors of Geometry edited by S Levy.
    • Computation of Hyperbolic Structures in Knot Theory contains a good overview of hyperbolic Dehn survey by J Weeks.

      Books.

    • Word processing in Groups by Epstein, D B A; Cannon, J W; Holt, D F; Levy, S V F; Paterson, M S; Thurston, W P.
    • J.P. Serre's book Trees.
    • Combinatorial Group Theory by Roger C Lyndon and Paul E Schupp.
    • Buildings: Theory and Applications by Abramenko, Peter and Brown, Kenneth S.
    • Topological Methods in Group Theory by Ross Geoghegan.
    • Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups by John Meier.
    • Topics in Geometric Group Theory by Pierre de la Harpe.
    • The Geometry and Topology of Coxeter Groups by Mike W Davis.
    • Metric Spaces of Non-Positive Curvature by Bridson, Martin R. and Haefliger, Andre.
    • Topics in Combinatorial Group Theory by Gilbert Baumslag.
    • Lectures on Coarse Geometry by J Roe.
    • Combinatorial Group Theory and Applications to Geometry by Collins, D.J., Grigorchuk, R.I., Kurchanov, P.F. and Zieschang, H.