Epimorphisms between Groups given by means of Generators and Defining Relations

Vella, Alfred Dominic (1977). Epimorphisms between Groups given by means of Generators and Defining Relations. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000deb8

Abstract

Chapter One is expository.

In Chapter Two we consider the following questions.
Let G = < X ; R >.
Is G Hopfian?
Is Aut(G) finitely generated?
Is there an algorithm to decide for any two words W₁,W₂ in X whether or not {W₁,W₂} generates G?
Let J be a set of group presentations. Is there an algorithm to decide whether or not two elements of J define isomorphic groups?

The study of these questions concerns the study of ependomorphisms between groups given by means of generators and defining relations. They have been considered by Pride in cases when G is a one-relator group with torsion.Using methods similar to those of Pride we obtain positive results for certain other groups particularly small cancellation groups. A problem related to Hopficity (stability) is also studied in Chapter Two.

In Chapter Three we study two questions. The first is. Let G=<X;R> and suppose that all elements of R can be expressed in the free group F on X as a set of freely reduced words T in a set of words W in X. Does sgp(W) have presentation < h ; T(h) > under the mapping h --> W?

We look at this question in detail and give many positive as well as negative results.

The other question in Chapter Three concerns malnormality. It is known that a subset of the generators of a one-relator group with torsion generates a malnormal subgroup.Is this result true for small cancellation groups? We show that the answer to this question is no in general but yes if all the relators are proper powers greater than two.