Classification of Infinite Primitive Jordan Permutation Groups

Abstract

We prove that every infinite primitive Jordan permutation group preserves a structure in one of a finite list of familiar families, or a limit of structures in one of these families. The structures are: semilinear orders (‘trees’) or betweenness relations induced from semilinear orders, chains of semilinear orders, points at infinity of a betweenness relation, linear and circular orders and the corresponding betweenness and separation relations, and Steiner systems. A Jordan group is a permutation group (G, Ω) such that there is a subset Γ ⊆ Ω satisfying various non-triviality assumptions, with G(Ω\Γ) transitive on Γ.