- 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,
isoperimetric inequalities of groups, finiteness properties of groups,
automatic and biautomatic groups, cubical complexes...
Here is a list of my papers and preprints
- There are good online resources for low-dimensional
topology and geometric group theory.
There are problem lists available at New York Group Theory Cooperative page, on Mladen Bestvina's page, and on Rob Kirby's page.
Jon McCammond maintains a Geometric Group Theory resource page. You can learn all about the geometric group theory (and low dimensional topology) community there; people, conferences, problem lists, journals etc.
Mark Brittenham maintains a Low Dimensional Topology page.
- I have three graduate students who are thinking about topics in
geometric group theory;
Eduardo Martinez, TaraLee Mecham, and Antara Mukherjee.
- Eduardo is currently organizing
the Student
Topology Seminar at OU.
- You can find out the current schedules of the OU Geometry and Topology Seminar and of the Geometric Group Theory Seminar.
Hyperbolic Groups
Free-by-Free Groups
| Free-by-cyclic groups are split extensions with free kernel and free quotient. The generators of the quotient act by (outer) automorphisms on the kernel. | |
| I am interested in the geometry of these groups; in particular in the existence of CAT(0) or CAT(-1) structures on these groups. In CAT(-1) structures for free-by-free groups with Andy Miller, we give examples of CAT(-1) free-by-free groups where the free quotient has rank 2. In Distortion of surface groups in CAT(0) free-by-cyclic groups Josh Barnard and I give examples of CAT(0) squared structures on a variety of free-by-cyclic groups. In CAT(0) and CAT(-1) dimensions of torsion free hyperbolic groups John Crisp and I give examples of hyperbolic free-by-cyclic groups which are not CAT(-1) in dimension 2, and examples which are not CAT(0) in dimension 2. These latter groups have since been shown (Shu and Wise) to be high dimensional CAT(0) cubical. In all three papers, we use combinatorial Morse theory to recognize that the fundamental group of a 2-complex is free-by-free (free-by-cyclic). |
| There are many interesting questions concerning the subgroup structure of free-by-cyclic groups. For example Feighn and Handel have proven that these groups are coherent (every finitely generated subgroup is finitely presented). Josh Barnard and I have given examples of CAT(0) free-by-cyclic groups which contain closed surface subgroups with polynomial and with exponential distortion. We produce hyperbolic free-by-cylic groups which contain exponentially distorted surface subgroups. The automorphisms in these examples are all reducible. Richard Weidmann has asked if there are free-by-cyclic groups with irreducible monodromy which contain exponentially distorted closed surface groups. |
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