Transitive primitive permutation groups acting on trees (Q1184184)

Let \(\Omega\) be a tree having no maximal elements. The automorphism group \(\text{Aut}(\Omega)\) and certain of its large subgroups \(G\) (those closed under ``patching'') are studied, building on work by \textit{M. Droste} [Mem. Am. Math. Soc. 334 (1981; Zbl 0574.06001)] and by \textit{M. Droste, W. C. Holland}, and \textit{H. D. Macpherson} [Proc. Lond. Math. Soc., III. Ser. 58, 454-478 (1989; Zbl 0636.20003)]. \((G,\Omega)\) is primitive if it has no proper congruences with convex classes. The transitive primitive groups \((G,\Omega)\) which are closed under patching are classified by examining their actions on the maximal subchains of \(\Omega\). If in addition \(\Omega\) is Dedekind complete, \((G,\Omega)\) must be either 2- homogeneous or else the right regular representation of the additive reals or integers. There is a nice array of examples.