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.
Published papers
(With Kim Ruane and Genevieve S. Walsh), The automorphism group of the free group of rank two is a CAT(0) group
To appear in Michigan Math. J.
We prove that the automorphism group of the braid group on four
strands acts faithfully and geometrically on a CAT(0) 2-complex.
This implies that the automorphism group of the free group of rank
two acts faithfully and geometrically on a CAT(0) 2-complex, in
contrast to the situation for rank three and above.
We show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.
Andrews--Curtis Groups and the Andrews--Curtis Conjecture
J. Group Theory 10 (2007), no. 3, 373--387; MR2320974
For an integer n at least two and a positive integer m, let AC(n,m) denote the group of Andrews–Curtis transformations of rank (n,m) and let F denote the free group of rank n + m. A subgroup AC(n,m) of Aut(F) is defined, and an anti-isomorphism AC(n,m) to AC(n,m) is described. We solve the generalized word problem for AC(n,m) in Aut(F) and discuss an associated reformulation of the Andrews–Curtis conjecture.
The manifestation of group ends in the Todd--Coxeter Coset Enumeration Procedure
Internat. J. Algebra Comput. 17 (2007), no. 1, 203--220; MR2300414
The issue of recognizing group properties, such as the cardinality of the group, directly from the dynamics of an incomplete coset enumeration is discussed. In particular, it is shown that the property of having two ends is recognizable in such a way. Further, sufficient conditions are given for termination of a coset enumeration with the declaration that the group under consideration has infinitely-many ends.
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.
Palindromic primitives and palindromic bases in the free group of rank two
The present paper records more details of the relationship between primitive elements and palindromes in F2, the free group of rank two. We characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F2. Further, we note that each cyclically reduced primitive element is either a palindrome, or the concatenation of two palindromes.
Slide show As given at AMS Sectional Meeting, University of New Hampshire, Durham, April 2006
Preprints submitted for publication
(With Mauricio Gutierrez and Kim Ruane), On the automorphisms of a graph product of abelian groups, August 2007, Updated March 2009
We study the automorphisms of a graph product of
finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut* W of Aut W, with Aut* W = Aut W whenever
vertex groups are finite and in a number of other cases.
We prove a number of structure results, including an informative semi-direct product decomposition of
Aut*W. We also give a number of applications, some of which are geometric in nature.
Slide show As given at the Spring Topology and Dynamics Conference, Milwaukee 2008
(With Kim Ruane), Normal forms for automorphisms of universal Coxeter groups and palindromic automorphisms of free groups, September 2009
We explicitly construct Markov languages of normal forms for the groups in the title of the paper and closely related groups. A Markov language of normal forms is a choice of "preferred spelling" for each group element such that the collection of choices is particularly simple in a language theoretic sense.
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 Danny Groves 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.
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.
