A fixed-point theorem for definably amenable groups
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Mathematical Logic
A fixed-point theorem for definably amenable groups Juan Felipe Carmona1
· Kevin Dávila1 · Alf Onshuus1 · Rafael Zamora2
Received: 18 August 2018 / Accepted: 6 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove an analogue of the fixed-point theorem for the case of definably amenable groups. Keywords Definable groups · Keisler measure · Fixed-point theorems · Definable amenability Mathematics Subject Classification 03C95 · 37B05 · 22F10
1 Introduction The notion of amenability was first introduced in the context of the Banach-Tarski paradox to identify when groups do not admit paradoxical decompositions; a discrete group is amenable if it admits a left-invariant finitely additive probability measure on all its subsets. This notion was then adapted by Rickert [5] to topological groups, restricting the domain of the measure to Borel subsets. In the context of locally compact groups (which includes the original Tarski context by taking the discrete topology) there are several properties that have been proved to be equivalent to amenability.1 Arguably, the most important one of those characterizations 1 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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Juan Felipe Carmona [email protected] Kevin Dávila [email protected] Alf Onshuus [email protected] Rafael Zamora [email protected]
1
Universidad de los Andes, Cra 1 No 1 8A-10, Bogotá, Colombia
2
Escuela de Matemática, Universidad de Costa Rica, San Pedro, San Jose, Costa Rica
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J. F. Carmona et al.
is the fixed-point property (a generalization of the Markov–Kakutani theorem), which states that any affine continuous action over a compact convex subset of a locally convex vector space has a fixed point. For a general topological group, this condition is weaker than amenability (in the sense we metioned), but it is still quite useful; in fact, this condition is called amenability in several functional analytic contexts, where most groups are not locally compact. The analogue of amenability for definable groups was defined in [2] as follows: A definable group is definably amenable if there is a left-invariant finitely-additive probability measure on all its definable subsets (a left-invariant global Keisler Measure). It is known that every stable group is definably amenable [4], and that groups definable in a dependent theory are definably amenable if and only if they have an f -generic type [3] (called strongly f -generic types in [1]). In this note we study definably amenable groups and prove that definable amenability is equivalent to a fixed-point condition. It is our hope that this could be a first step in finding fixed-point conditions which could eventually lead to a more general “amenability” notion for definable groups, in much the same way it happens for non-locally compact topological groups. The paper is divided as follows: in Sect. 2. we study the σ -
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