Countable 1-transitive trees

Chicot, Katie M. and Truss, John K. (2017). Countable 1-transitive trees. In: Droste, Manfred; Fuchs, László; Goldsmith, Brendan and Strüngmann, Lutz eds. Groups, Modules and Model Theory - Surveys and Recent Developments, in Memory of Rüdiger Göbel. Springer International Publishing AG, pp. 225–268.

DOI: https://doi.org/10.1007/978-3-319-51718-6_11

Abstract

We give a survey of three pieces of work, on 2-transitive trees (Droste, Memoirs Am Math Soc 57(334) 1985), on weakly 2-transitive trees (Droste et al., Proc Lond Math Soc 58:454–494, 1989), and on lower 1-transitive linear orders (Barbina and Chicot, Towards a classification of the countable 1-transitive trees: countable lower 1-transitive linear orders. arXiv:1504.03372), all in the countable case. We lead on from these to give a complete description of all the countable 1-transitive trees. In fact the work of Barbina and Chicot (Towards a classification of the countable 1-transitive trees: countable lower 1-transitive linear orders. arXiv:1504.03372) was carried out as a preliminary to finding such a description. This is because the maximal chains in any 1-transitive tree are easily seen to be lower 1-transitive, but are not necessarily 1-transitive. In fact a more involved set-up has to be considered, namely a coloured version of the same situation (where ‘colours’ correspond to various types of ramification point), so a major part of what we do here is to describe a large class of countable coloured lower 1-transitive linear orders and go on to use this to complete the description of all countable 1-transitive trees. This final stage involves analysing how the possible coloured branches can fit together, with particular attention to the possibilities for cones at ramification points.

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