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A short note on extreme amenability of automorphism groups of \(\omega\)-categorical structures. (English) Zbl 1341.06001

A topological group is extremely amenable if every continuous action \(\cdot\colon G\times X\to X\) on a compact Hausdorff space \(X\) has a joint fixpoint, that means that there is an \(x_0\in X\) such that \(g\cdot x_0=x_0\) for all \(g\in G\).
Here the author shows that the automorphism group of every \(\omega\)-categorical linear order is extremely amenable. This result implies that an oligomorphic permutation group \(G\) is contained in an extremely amenable permutation group if and only if it preserves a linear order. The paper is based on F. G. Dorais et al. [Order 30, No. 2, 415-426 (2013; Zbl 1279.06002)].

MSC:

06A05 Total orders
03C35 Categoricity and completeness of theories
20B27 Infinite automorphism groups
43A07 Means on groups, semigroups, etc.; amenable groups
03C15 Model theory of denumerable and separable structures
03C50 Models with special properties (saturated, rigid, etc.)
05C55 Generalized Ramsey theory
20B07 General theory for infinite permutation groups

Citations:

Zbl 1279.06002
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References:

[1] Cameron, P.J.: Oligomorphic Permutation Groups. Cambridge University Press, Cambridge (1990) · Zbl 0813.20002
[2] Dorais, F.G., Gubkin, S., McDonald, D., Rivera, M.: Automorphism groups of countably categorical linear orders are extremely amenable. Order 30, 415-426 (2013) · Zbl 1279.06002
[3] Fremlin, D.H.: Measure Theory: Topological Measure Spaces, vol. 4. Torres Fremlin (2003) · Zbl 0273.46035
[4] Kechris, A.S., Pestov, V.G., Todorčević, S.: Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15, 106-189 (2005) · Zbl 1084.54014
[5] Pestov, V.G.: On free actions, minimal flows and a problem by Ellis. Trans. Am. Math. Soc. 350, 4149-4165 (1998) · Zbl 0911.54034
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