Sa. Adeleke et D. Macpherson, CLASSIFICATION OF INFINITE PRIMITIVE JORDAN PERMUTATION-GROUPS, Proceedings of the London Mathematical Society, 72, 1996, pp. 63-123
We prove that every infinite primitive Jordan permutation group preser
ves a structure in one of a finite list of familiar families, or a lim
it of structures in one of these families. The structures are: semilin
ear orders ('trees') or betweenness relations induced from semilinear
orders, chains of semilinear orders, points at infinity of a betweenne
ss relation, linear and circular orders and the corresponding betweenn
ess and separation relations, and Steiner systems. A Jordan group is a
permutation group (G, Omega) such that there is a subset Gamma subset
of or equal to Omega satisfying various non-triviality assumptions, w
ith G((Omega/Gamma)) transitive on Gamma.