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 gamma_n(G) and gamma_n(H) are elementarily equivalent, as are the quotient groups G/gamma_n(G) and G/gamma_n(H). We go on to consider those groups universally equivalent to F_2(VN_c), where the free groups of the variety V 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 V_k · · ·V_1, where V_i 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:432573 |
| Date | January 2006 |
| Creators | Wharton, Elizabeth |
| Contributors | Wilson, John |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:7bd8d05b-4ff6-4326-8463-f896e2862e25 |
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