Automorphism groups of countable structures and groups of measurable functions

Autor: Aleksandra Kwiatkowska, Maciej Malicki
Rok vydání: 2019
Předmět:
Zdroj: Israel Journal of Mathematics. 230:335-360
ISSN: 1565-8511
0021-2172
DOI: 10.1007/s11856-018-1825-7
Popis: Let G be a topological group and let μ be the Lebesgue measure on the interval [0, 1]. We let L0(G) be the topological group of all μ-equivalence classes of μ-measurable functions defined on [0, 1] with values in G, taken with the pointwise multiplication and the topology of convergence in measure. We show that for a Polish group G, if L0(G) has ample generics, then G has ample generics, thus the converse to a result of Kaichouh and Le Maitre. We further study topological similarity classes and conjugacy classes for many groups Aut(M) and L0(Aut(M)), where M is a countable structure. We make a connection between the structure of groups generated by tuples, the Hrushovski property, and the structure of their topological similarity classes. In particular, we prove the trichotomy that for every tuple $$\bar f$$ of Aut(M), where M is a countable structure such that algebraic closures of finite sets are finite, either the countable group $$\langle \bar f \rangle$$ is precompact, or it is discrete, or the similarity class of $$\bar f$$ is meager, in particular the conjugacy class of $$\bar f$$ is meager. We prove an analogous trichotomy for groups L0(Aut(M)).
Databáze: OpenAIRE
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