The model theory of certain infinite soluble groups

Abstract

This thesis is concerned with aspects of the model theory of infinite soluble groups. The results proved lie on the border between group theory and model theory: the questions asked are of a model-theoretic nature but the techniques used are mainly group-theoretic in character.

We present a characterization of those groups contained in the universal closure of a restricted wreath product U wr G, where U is an abelian group of zero or finite square-free exponent and G is a torsion-free soluble group with a bound on the class of its nilpotent subgroups. For certain choices of G we are able to use this characterization to prove further results about these groups; in particular, results related to the decidability of their universal theories.

The latter part of this work consists of a number of independent but related topics.

We show that if G is a finitely generated abelian by-metanilpotent group and H is elementarily equivalent to G then the subgroups and are elementarily equivalent, as are the quotient groups and .

We go on to consider those groups universally equivalent to , where the free groups of the variety are residually finite p-groups for infinitely many primes p, distinguishing between the cases when c = 1 and when c > 2.

Finally, we address some important questions concerning the theories of free groups in product varieties , where is a nilpotent variety whose free groups are torsion-free; in particular we address questions about the decidability of the elementary and universal theories of such groups. Results mentioned in both of the previous two
paragraphs have applications here.