Descriptive Set Theory and Dynamics of Countable Groups

Citation

Shinko, Forte (2022) Descriptive Set Theory and Dynamics of Countable Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/egch-kp69. https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796

Abstract

This thesis comprises four papers.

1. We show that for any Polish group G and any countable normal subgroup Γ ⊳ G, the coset equivalence relation G/Γ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.

2. Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.

3. Let Γ be a countable group. A classical theorem of Thorisson states that if X is a standard Borel Γ-space and µ and ν are Borel probability measures on X which agree on every Γ-invariant subset, then µ and ν are equidecomposable, i.e., there are Borel measures (µ_γ)_γϵΓ on X such that µ = Σ_γµ_γ and ν = Σ_γγµ_γ. We establish a generalization of this result to cardinal algebras.

4. Let R be a ring equipped with a proper norm. We show that under suitable conditions on R, there is a natural basis under continuous linear injection for the set of Polish R-modules which are not countably generated. When R is a division ring, this basis can be taken to be a singleton.

Thesis Files