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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. If you would like a copy of one of these papers, send me an email and I will be happy to oblige.

- Pacific J. Math., 284-1 (2016), pp. 41-77; MR3530862
- We consider the question of determining whether or not a given group (especially one generated by involutions) is a right-angled Coxeter group.
We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization
to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. As a corollary, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover an existing result stating that if G satisfies a particular graph condition (called no SILs), then Aut
^{0}(W) is a right-angled Coxeter group.

- We consider the question of determining whether or not a given group (especially one generated by involutions) is a right-angled Coxeter group.
We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization
to describe a process for constructing candidate right-angled Coxeter presentations for a given group or proving that one cannot exist. As a corollary, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover an existing result stating that if G satisfies a particular graph condition (called no SILs), then Aut

- Topology Proc., 48 (2016), pp 277-287. (This paper was e-published on December 3, 2015.); MR3431824
- We show that any split extension of a right-angled Coxeter group W by a generating automorphism of finite order acts faithfully and geometrically on a CAT(0) metric space.

- Bull. Aust. Math. Soc., 91 (2015), no. 3, pp 426-434; MR3338967
- We prove that the groups presented by finite convergent monadic rewriting systems with generators of finite order are exactly the free products of finitely many finite groups, thereby confirming Gilman’s Conjecture in a special case. We also prove that the finite cyclic groups of order at least three are the only finite groups admitting a presentation by more than one finite convergent monadic rewriting system (up to relabeling), and these admit presentation by exactly two such rewriting systems.

- Illinois J. Math., 58 (2014), no. 1, pp 27-41; MR3331840
- We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.

- Algebra Discrete Math., 14 (2012), no. 2, pp. 239-266; MR3099973
- We prove that if W is the free product of at least four groups of order 2, then the automorphism group of the McCullough-Miller space corresponding to W is isomorphic to group of outer automorphisms of W. We also prove that, for each integer n such that n > 2, the automorphism group of the hypertree complex of rank n is isomorphic to the symmetric group of rank n.
- The preprint

- Comm. Algebra, 40 (2012), no. 3, 1142-1150; MR2899931
- We characterize certain properties of the derived series of Coxeter groups by properties of the corresponding Coxeter graphs. In particular, we give necessary and sufficient conditions for a Coxeter group to be quasiperfect.

- Groups Geom. Dyn. 6 (2012), no. 1, pp. 125-153; MR2888948
- 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

- We study the automorphisms of a graph product of
finitely-generated abelian groups W. More precisely, we study a natural subgroup Aut

- Internat. J. Algebra Comput. 20 (2010), no. 8, pp. 1063-1086; MR2747416
- 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.

- Michigan Math. J. 59 (2010), 297-302; MR2677622
- 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.

- Bull. Aust. Math. Soc. 77 (2008), no. 2, 187-196; MR2428781
- 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 T
_{0}property. Our results build on results by Droms, Laurence and Radcliffe.

- 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 T

- 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. - Slide show

- For an integer n at least two and a positive integer m, let

- 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: For Theorem 3 only, 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.

- J. Algebra 304 (2006), no. 1, 359-366; MR2256396
- The present paper records more details of the relationship between primitive elements and palindromes in F
_{2}, 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 F_{2}. 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

- The present paper records more details of the relationship between primitive elements and palindromes in F

- Following the advice of the editor and referee, this paper has been undergoing a major rewrite. It will be available again soon.

- My thesis was supervised by Martin Bridson and Danny Groves and Marc Lackenby. It was submitted at the University of Oxford, 2004.

- 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.