• Corpus ID: 120671793

Jordan groups and homogeneous structures

@inproceedings{BradleyWilliams2014JordanGA,
  title={Jordan groups and homogeneous structures},
  author={David Bradley-Williams},
  year={2014}
}

• David Bradley-Williams
• Published 1 November 2014
• Mathematics

A permutation group G acting transitively on a set Ω is a Jordan group if there is a proper subset Γ ⊂ Ω, subject to some non-triviality conditions, such that the pointwise stabiliser in G of Ω \ Γ is transitive on Γ. Such sets Γ are called Jordan sets for G. Here we study infinite primitive Jordan groups which are automorphism groups of first order relational structures. We find a model theoretic application in classifying the reducts of an infinite family of semilinearly ordered partial orders, and apply a model-theoretic construction technique to obtain examples of Jordan groups. The kinds of structures preserved by an infinite primitive Jordan group with a primitive Jordan set were classified by Adeleke and Neumann [4]. In Chapter 2 we apply this classification and further results on the combinatorial behaviour of the families of primitive Jordan sets to obtain an infinite family of non-conjugate, maximal closed subgroups of Sym(ω). We obtain a classification of the reducts (up to first order interdefinability) of an infinite family of semilinearly ordered trees. The classification of Jordan groups was continued by Adeleke and Macpherson [2], and they determined the kinds of structures preserved by infinite primitive Jordan groups, without the assumption that there is a primitive Jordan set. This list includes some exotic kinds of relational structures, including so-called ‘limits of betweenness relations’. Nonisomorphic examples of infinite primitive Jordan groups preserving a limit of betweenness relations have been constructed by Adeleke [1] and Bhattacharjee and Macpherson [6]. In Chapter 3 we develop the construction of Bhattacharjee and Macpherson to construct what we believe to be a limit of betweenness relations preserved by the group constructed by Adeleke.