- Department of Mathematics
Professor and Chair
Bromfield-Pearson, Room 103
To understand what I study, it is important to understand how I came to study the subject. I was originally recruited to graduate school to work on a project with Professor DeWitt Sumners that involved applications of topology to DNA research. The idea is that biologists can take pictures of DNA using a special microscope but once they have the pictures, they have to be analyzed. Such a picture is a two dimensional projection of a configuration of strands in three dimensional space. It is a well developed theory of mathematics called knot theory which studies such configurations. In particular, the question of whether two such configurations are the same is a difficult math problem. Imagine taking just one very long piece of string that has been bunched up very tight. If you pull the two ends of the string, you could end up with a straight piece of string or one with a big knot in the middle. How do you know which one of these you will get only looking at the original bunched up string? To a given configuration of strands, one can associate a mathematical object called a group. It is an important theorem that says two configurations are the same if and only if they have the same associated group. So while studying under Professor Sumners, I encountered the subject of group theory and became completely fascinated with this subject on its own.
The concept of a group arises quite naturally in the study of geometric objects. The set of symmetries (or rigid motions) of a geometric object form a group because if you do two or more of these in succession, you get another. For example, there are 8 symmetries of the square but all 8 can be constructed as a combination of a rotation by 90 degrees and a flip through the center axis. Thus you only have to know these two to understand all 8. Likewise, any rigid motion of the standard Euclidean plane can be constructed using a combination of horizontal translation and rotation about the origin. These two motions allow you to describe the infinitely many rigid motions of the plane. Computing the symmetry group of two geometric objects can also distinguish between the objects as in the case of the knots. If the groups are different, then the spaces are different. Showing two groups are different is sometimes an easier problem than directly showing the spaces are different.
During the second summer of graduate school, I took a course on fuchsian groups. These are groups of rigid motions of the non-euclidean (or hyperbolic) plane. In this course, we used a combination of geometry, group theory, and topology which were my favorite subjects from my elementary graduate work. I asked the Professor if he could suggest some research papers for me to read which would take me beyond the material in the course. He gave me several articles which outlined a new approach to studying a class of groups that are similar to fuchsian groups only much more general. In particular, this class includes many of the knot groups that first got me to graduate school in the first place! This area of research is called Geometric Group Theory and it has been a very popular and productive area of research for the past 25-30 years. As with many areas of mathematics, the beauty of the subject comes from many simple ideas from different disciplines woven together to reveal powerful new tools for solving problems.
Geometric Group Theory/Topology
Selected Publications and Presentations
The automorphism group of the free group of rank two is a CAT(0) group
In this paper, we prove that the automorphism groupof the braid group on 4 strands acts faithfully and geometrically on a CAT(0) 2-complex. This is used to show that Aut(F_2) also acts faithfully and geometrically on a CAT(0) space because these two groups are isomorphic. This a joint paper with A. Piggott.and G. Walsh.
The automorphism group of a graph product with no SIL
In this paper, we study the subgroup of automorphisms of a graph product of cyclic groups that is generated by the partial conjugations. This subgroup is itself a graph product of cyclic groups provided the defining graph has no SIL. In particular, if the vertex groups are all finite cyclic, one can conclude that this (finite index) subgroup of the automorphism group is CAT(0). This a joint paper with R. Charney, N.Stambaugh, A. Vijayan.
Normal forms for automorphisms of universal Coxeter groups and palindromic automorphisms of free groups
In this paper, we explicitly construct Markov languages of normal forms for the groups in the title of this paper. This a joint paper with A. Piggott.
On the automorphisms of a graph product of abelian groups
In this paper, we consider investigate the structure of the automorphism group of a graph product of abelian groups. In particular, this includes right-angled Artin and Coxeter groups. This a joint paper with A. Piggott and M. Gutierrez.
CAT(0) groups with specified boundary
In this paper, I consider the question of whether the homeomorphism type of the visual boundary determines the space and/or the group for some first examples.
CAT(0) boundary of truncated hyperbolic space
In this paper, I compute the CAT(0) boundary of truncated hyperbolic space.
Some geometric groups with rapid decay
In this paper, Indira Chatterji and I show that any lattice in a rank one Lie group satisfies the Baum-Connes Conjecture.
Local connectivity of right-angled Coxeter group boundaries
We give conditions on the defining graph of a right-angled Coxeter group that guarantee any CAT(0) boundary of the group must be locally connected.
CAT(0) HNN-extensions with non-locally connected boundary
We show how to construct CAT(0) groups with non-locally connected boundary using HNN-extensions.
Amalgamated products with non-locally connected boundary
CAT(0) We show how to construct CAT(0) groups with non-locally connecte boundary using amalgamated products. In particular, we show how to construct one-ended right-angled Coxeter groups where the Davis complex has non-locally connected boundary simply by giving conditions on the defining graph of the group.
Dynamics of the group action on the boundary of a CAT(0) space
We investigate how an individual hyperbolic isometry in a CAT(0) group must act on the boundary of the CAT(0) space.
Boundaries of groups of the form GxH
We prove that groups of the form GxH where G and H are both hyperbolic have unique CAT(0) boundary.
The angle question
We investigate CAT(0) groups which contain an infinite order in the center. Without loss of generality, one can assume the group is of the form GxZ. We show that although G does not have to be quasi-convex in the space, there is a well-defined angle which G makes with the central element the comes from the action.