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Automorphism groups of countably categorical linear orders are extremely amenable. (English) Zbl 1279.06002
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.

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
Full Text: DOI arXiv
[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
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