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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.

##### 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
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##### References:
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