Abstract
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
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Notes
In this paper we say that a topological group is amenable if it admits a left invariant finitely additive probability measure on the Borel subsets.
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Acknowledgements
We would like to thank the referee for their numerous suggestions.
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Appendix
Appendix
1.1 Definability and \(\sigma \)-continuity
The notion of \(\sigma \)-continuity is very close to the one of definable map. In this paper we did not not make use of the latter, however it is worth to establish more precisely the relation between them.
Definition 8
Let Y be a definable set in a structure M and C a compact space. A map \(f:Y\rightarrow C\) is definable if for every disjoint closed sets \(C_1\) and \(C_2\) there exists \(Y'\) definable such that \(f^{-1}(C_1)\subset Y'\) and \(Y'\cap f^{-1}(C_2)=\emptyset \).
From the proof of Theorem 4 we have the following characterization of definable functions:
Corollary 1
Let X be definable in a structure M, C be a compact space, and \(f : X \rightarrow C\). Then we have the following properties.
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1.
Suppose M is \(\omega _1\)-saturated. If f is \(\sigma \)-continuous, then f is definable.
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2.
Suppose C is second-countable. If f is definable, then f is \(\sigma \)-continuous.
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Carmona, J.F., Dávila, K., Onshuus, A. et al. A fixed-point theorem for definably amenable groups. Arch. Math. Logic 60, 413–424 (2021). https://doi.org/10.1007/s00153-020-00748-1
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DOI: https://doi.org/10.1007/s00153-020-00748-1