Dugald Macpherson (Leeds)

Jordan permutation groups and limits of treelike structures

A transitive permutation group $G$ on a set $X$ is a Jordan group if there is a subset $A$ of $X$ (a 'Jordan set') with $|A|>1$ such that the subgroup of $G$ fixing the complement of $A$ is transitive on $A$ (+ a non-degeneracy condition that if $G$ is $k$-transitive on $X$ then $|X \setminus A|\geq k$.) So if $X$ carries some first-order structure, this is a bit like saying all elements of $A$ realise the same type over $X \setminus A$. Work of Adeleke, myself and Neumann in the 1990s gave a kind of classification of Jordan groups which are 'primitive', i.e. preserve no proper non-trivial equivalence relation on $X$. Many key examples can be seen as Fraïssé limits.

I will discuss examples, and also sketch recent work in Asma Almazaydeh's thesis (and subsequent work with her) on certain mysterious $\omega$-categorical structures which are limits of treelike structures. This relates to earlier work with Meenaxi Bhattacharjee, and to a recent preprint of David Bradley-Williams and John Truss.

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