Mašulović, Dragan A short note on extreme amenability of automorphism groups of \(\omega\)-categorical structures. (English) Zbl 1341.06001 Order 33, No. 1, 67-70 (2016). 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)]. Reviewer: Martin Weese (Potsdam) Cited in 1 Document 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 Keywords:extremely amenable groups; omega-categorical structures; automorphism groups Citations:Zbl 1279.06002 PDFBibTeX XMLCite \textit{D. Mašulović}, Order 33, No. 1, 67--70 (2016; Zbl 1341.06001) Full Text: DOI 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 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.