Research Page

I am interested in geometric group theory and low dimensional topology. Specific topics include hyperbolic groups (and their subgroups), Artin groups, CAT(0) groups, Morse theory, (high dimensional) filling invariants of groups, finiteness properties of groups, automatic and biautomatic groups, cubical complexes...
Here is a list of my papers and preprints.
Graduate Students.
- I have one graduate student who is currently thinking about topics in geometric group theory; Ignat Soroko.
- 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 now a member of the mathematics department at Memorial University of Newfoundland.
- In Fall 2008 Antara Mukherjee completed her PhD thesis on second order Dehn functions of Nil and Sol lattices and Varopolous transport. She is one of the mathematics faculty at the Citadel in Charlestown, South Carolina.
- In Summer 2009 TaraLee Mecham completed her PhD thesis on largeness of graphs of abelian groups. After graduation TaraLee joined the mathematics department at the College of Mount St Joseph in Cincinnati, Ohio. She was a Project NExT fellow.
- In Summer 2012 Sang Rae Lee completed his thesis on cube complex structures associated to Houghton's groups. He is currently a postdoctoral associate at Texas A&M University.
Consider applying for a post-doctoral appointment at the University of Oklahoma. Here are our current and past geometric group theory post-doctoral associates.
- Josh Barnard (2004-2007). Currently at the University of South Alabama.
- Pallavi Dani (2005-2008). Currently at Louisiana State University.
- Matt Clay (2006-2009). Currently at Allegheny College.
- Dan Guralnik (2008-2011). Dan is currently an AFOSR MURI Postdoc at the Electric and Systems Enginnering department, University of Pennsylvania.
Current schedules of the OU Geometry and Topology Seminar and the Student Topology and Geometry Seminar.
If you are interested in learning about geometric group theory, here are some good online resources.
- The Wikipedia Geometric Group Theory page is a good place to start for an overview.
- There are problem lists available at the Geometric Group Theory Wiki, the New York Group Theory Cooperative, Mladen Bestvina's page, and Rob Kirby's page.
- Jon McCammond maintains a Geometric Group Theory resource page. Mark Brittenham maintains a Low Dimensional Topology resource page. Use these pages to find out about the geometric group theory and low dimensional topology community: people; conferences; problem lists; journals; software; etc.
- There are a number of blogs about low-dimensional topology and geometric group theory.
  - Low Dimensional Topology maintained by Nathan Dunfield, Jesse Johnson, Daniel Moskovich, Henry Wilton and perhaps others.
  - Here there be dragons. The Berstein Seminar Blog maintained by Tim Riley.
  - Sketches of Topology maintained by Kenneth Baker. Beautiful stuff.
  - Geometry and the Imagination maintained by Danny Calegari.
  - Geometric Group Theory Past course blog mainted by Henry Wilton.
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.
- It is important to have a solid foundation in algebraic topology, particularly fundamental groups and covering spaces.
  Some Historical Results/Development.
- In 1911 Max Dehn first formulated the word problem, conjugacy problem and the isomorphism problem for finitely presented groups, and gave a wonderfully elegant solution (now known as Dehn's algorithm) to the word problem of the fundamental groups of closed surfaces of genus at least 2.
- 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. IHÃS 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.
- Karen Vogtmann's Notices article gives a good introduction to the study of outer automorphisms of free groups. It includes references to several excellent survey articles on this subject.
- Mladen Bestvina has some introductory and survey papers on a variety of topics. The topology of Out(F_n), R-trees in topology, geometry, and group theory, and PL Morse theory.
  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.