Andrew J. Duncan

• Newcastle University

In this note we give two examples of partially commutative subgroups of partially commutative groups. Our examples are counterexamples to the Extension Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda, \cite{KK}. On the other hand we extend the class of partially commutative groups for which it is known that the Extension Gr...

Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregro...

We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition...

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at o...

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at o...

Elements of the commutator subgroup of a free group can be presented as values of canonical forms, called Wicks forms. We show that, starting from sufficiently high genus g, there is a sequence of words w(g) which can be presented by f(g) distinct Wicks forms, where f(g)>g!. Moreover we may choose these words w(g) to be square free.

In 1971 J. Stallings introduced a generalisation of amalgamated products of groups – called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be...

The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct an arithmetic subgroup $St(L(G))$, represented as a subgroup of $GL(n,Z)$, where $n$ is the number of vertices of the graph $\Gamma$. In th...

We adapt the Deutsch-Josza algorithm to the context of formal language theory. Specifically, we use the algorithm to distinguish between trivial and nontrivial words in groups given by finite presentations, under the promise that a word is of a certain type. This is done by extending the original algorithm to functions of arbitrary length binary ou...

To a finite graph there corresponds a free partially commutative group: with the given graph as commutation graph. In this paper we construct an orthogonality theory for graphs and their corresponding free partially commutative groups. The theory developed here provides tools for the study of the structure of partially commutative groups, their uni...

Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially commutative groups (or right-angled Artin groups). In this paper we construct orthogonality theory for graphs with th...

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugat...

We prove results that will be required for the study of the algebraic geometry of partially commutative groups. We define classes of groups axiomatized by sentences determined by a graph. Among the classes which arise this way we find CSA- and CT-groups. We study the centralisers of a group, with particular attention to the height of the lattice of...

In a previous paper we investigated the centraliser dimension of groups. In the current paper we study properties of centraliser dimension for the class of free partially commutative groups and, as a corollary, we obtain an efficient algorithm for computation of centraliser dimension in these groups.

Exponential equations in free groups were studied initially by Lyndon and Schutzenberger and then by Comerford and Edmunds. Comerford and Edmunds showed that the problem of determining whether or not the class of quadratic exponential equations have solution is decidable, in finitely generated free groups. In this paper we show that for finite syst...

In a previous paper we investigated the centraliser dimension of groups. In the current paper we study properties of centraliser dimension for the class of free partially commutative groups and, as a corollary, we obtain an efficient algorithm for computation of centraliser dimension in these groups.

In this paper we establish results that will be required for the study of the algebraic geometry of partially-commutative groups. We define classes of groups axiomatised by sentences determined by a graph. Among the classes which arise this way we find CSA and CT groups. We study the centraliser dimension of a group, with particular attention to th...

Hoyer has given a generalisation of the Deutsch--Jozsa algorithm which uses the Fourier transform on a group G which is (in general) non-Abelian. His algorithm distinguishes between functions which are either perfectly balanced (m-to-one) or constant, with certainty, and using a single quantum query. Here, we show that this algorithm (which we call...

We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.

The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups...

W. A. Bogley and M. A. Gutierrez [ 2 ] have recently obtained an eight-term exact homology sequence that relates the integral homology of a quotient group Г/ MN , where M and N are normal subgroups of the group Г, to the integral homology of the free product Г/ M * Г/ N in dimensions ≤3 by means of connecting terms constructed from commutator subgr...

Spelling theorems, a Cohen-Lyndon theorem and a Magnus theorem are proved for one-relator products of arbitrary groups, in cases where the relator is a sufficiently high power.

Following a conjecture of Weinbaum, we show that every nonperiodic word W of length at least 2 in a free group has a cyclic permutation of the form U V, where each of U and V occur precisely once as a cyclic subword of W and neither occurs as a cyclic subword of W-1. In fact, we prove a somewhat stronger version of this result and also give a numbe...

Following a conjecture of Weinbaum, we show that every nonperiodic word W of length at least 2 in a free group has a cyclic permutation of the form UV, where each of U and V occur precisely once as a cyclic subword of W and neither occurs as a cyclic subword of W -1 . In fact, we prove a somewhat stronger version of this result and also give a numb...

This paper is devoted to the study of unions of ideals in commutative rings. The starting point is the prime avoidance lemma and an accompanying but diverse body of results on coverings of ideals by unions of ideals, which is described in Section 1. In Sections 4 and 5 these known facts, about finite and infinite unions, are combined and generalize...

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