My research is in Combinatorial and Geometric Group Theory,
with a particular focus on the automorphism groups of groups and
algorithms in group theory. You can find preprints, seminar slides and other things related to my research below.
Errata: One should add the hypothesis that each finite presentation we deal with has a generator which also appears as a relation and does not appear in any other relation. If your presentation does not have such a generator, simply add one with a Tietze transformation.
Slide show As given at College Station, November 2005
Andrews--Curtis Groups and the Andrews--Curtis Conjecture
J. Group Theory 10 (2007), no. 3, 373--387; MR2320974
Slide show As given at the Spring Topology and Dynamics Conference, Milwaukee 2008
Under revision
Detecting the growth of free-group automorphisms by their action on the homology of subgroups of finite index
Following the advice of the editor and referee, this paper has been undergoing a
major rewrite. It will be available again soon.
D.Phil. Thesis
The topology of finite graphs, recognition and the growth of free-group automorphisms
My thesis was supervised by Martin Bridson and Marc Lackenby. It was submitted at the University of Oxford, 2004.
A letter
A letter to the Seminario Teoria de Grupos de la Universidad de los Andes, August 27, 2007
My friend and collaborator Mauricio Gutierrez asked me to record some
thoughts to provide a discussion point for the seminar in the title of the letter.
The letter contains a sketch argument for the rigidity of
right-angled Coxeter groups, a result previously proved
(in greater generality) by Radcliffe, Laurence, Bahls and others.
How Michael met Jessica: romance and error detecting codes
Slides and associated spreadsheet from a seminar given at Bowdoin College, October 2007. In this seminar
we describe an example of an error detecting code, the ISBN number of a book, as an immediate application of
elementary number theory.
Download some coset enumeration software designed and implemented by students at Tufts University for the course COMP190 in Spring 2006. The software is called
"Todd--Coxeter Enumeration Using Graphs", and is named after a paper by Stallings and Wolf (1987). The software demonstrates a graphical interpretation, due to Stallings and Wolf, of the Todd--Coxeter Coset Enumeration Procedure. In my opinion, this interpretation possesses a conceptual clarity well-suited to the classroom. Download, install and enjoy. If you use this software, especially the source code, please acknowledge the students who wrote it (their names can be found in the documentation) and Tufts University. When using the software, you must include a generator which is also a relation and does not appear in any other relation (see the Errata above).
This program allows the user to enter words in {x, y, X, Y}*
and then tests to see if the word is a reduced primitive in F(x, y).
The algorithm for testing cyclically reduced primitive elements
is very fast (linear in length of input)
and follows from Osborne and Zieshang (Invent. Math, 1981). Of course, Whitehead's Algorithm is also very fast and works for higher rank free groups, but this is easy to do by hand and fun to play with.
