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.

Viewing alternatives

Metrics

Item Actions

Export

You can export this page using these formats Digital Object Identifier - DOI