Dorais, François Gilbert; Gubkin, Steven; McDonald, Daniel; Rivera, Manuel Automorphism groups of countably categorical linear orders are extremely amenable. (English) Zbl 1279.06002 Order 30, No. 2, 415-426 (2013). A topological group \(G\) is extremely amenable if every action of \(G\) on a compact Hausdorff space has a fixed point. It was shown by V. G. Pestov [Trans. Am. Math. Soc. 350, No. 10, 4149–4165 (1998; Zbl 0911.54034)] that the automorphism group of the linear order of \(\mathbb{Q}\) is extremely amenable.Here the authors show that every countable linear order has a canonical sum-shuffle expression. They use Fraïssé limits to show that every countably categorical linear order is extremely amenable. They use methods of A. S. Kechris et al. [Geom. Funct. Anal. 15, No. 1, 106–189 (2005; Zbl 1084.54014)] to derive structural Ramsey theorems. Reviewer: Martin Weese (Potsdam) Cited in 1 ReviewCited in 1 Document MSC: 06A05 Total orders 03C35 Categoricity and completeness of theories 05C55 Generalized Ramsey theory 20B27 Infinite automorphism groups 43A07 Means on groups, semigroups, etc.; amenable groups Keywords:linear orders; automorphism groups; countable categoricity; extreme amenability; Fraïssé classes; Ramsey property; topological group; sum-shuffle expression PDF BibTeX XML Cite \textit{F. G. Dorais} et al., Order 30, No. 2, 415--426 (2013; Zbl 1279.06002) Full Text: DOI arXiv References: [1] Fraïssé, R, Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Ecole Norm. Sup., 71, 363-388, (1954) · Zbl 0057.04206 [2] Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997) · Zbl 0873.03036 [3] Kechris, AS; Pestov, VG; Todorcevic, S, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups, Geom. Funct. Anal., 15, 106-189, (2005) · Zbl 1084.54014 [4] Nguyen Van The, L, Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces and dynamics of their isometry groups, Eur. J. Comb., 30, 934-945, (2009) · Zbl 1171.05436 [5] Pestov, VG, On free actions, minimal flows, and a problem by Ellis, Trans. Am. Math. Soc., 350, 4149-4165, (1998) · Zbl 0911.54034 [6] Rado, R, Direct decomposition of partitions, J. Lond. Math. Soc., 29, 71-83, (1954) · Zbl 0055.04903 [7] Ramsey, FP, On a problem of formal logic, Proc. Lond. Math. Soc., 30, 264-286, (1930) · JFM 55.0032.04 [8] Rosenstein, JG, \(ℵ \sb{0}\)-categoricity of linear orderings, Fund. Math., 64, 1-5, (1969) · Zbl 0179.01303 [9] Rosenstein, JG, Linear orderings, No. 98, (1982), London · Zbl 0488.04002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.