Adam Piggott

On the automorphisms of a graph product of abelian groups

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 a semi-direct product decomposition Aut* W = (Inn W x Out(0) W) x Aut(1) W. We also give a number of applications, some of which are geometric in nature.

Palindromic primitives and palindromic bases in the free group of rank two

Abstract 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.

RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

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 To property. Our results build on results by Droms, Laurence and Radcliffe.

The automorphism group of the free group of rank 2 is a CAT(0) group

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.

Recognizing right-angled Coxeter groups using involutions

We consider the question of determining whether 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. We provide some first applications. In addition, we provide an elementary proof of rigidity of the defining graph for a right-angled Coxeter group. We also recover a result stating that if the defining graph contains no SILs, then Aut^0(W) is a right-angled Coxeter group.

Andrews–Curtis groups and the Andrews–Curtis conjecture

Abstract For an integer n at least two and a positive integer m, let C(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 C(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.

On groups presented by monadic rewriting systems with generators of finite order

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 Gilmans 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 relabelling), and these admit presentation by exactly two such rewriting systems.

On the Derived Length of Coxeter Groups

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.

THE MANIFESTATION OF GROUP ENDS IN THE TODD–COXETER COSET ENUMERATION PROCEDURE

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.