Descriptive Set Theory and Dynamics of Countable Groups

Author: Shinko, Forte

Year: 2022

Degree: Dissertation (Ph.D.)

Advisor: Kechris, Alexander S.

Committee Members: Tamuz, Omer; Conlon, David; Vidnyánszky, Zoltán; Kechris, Alexander S.

Option: Mathematics

DOI: 10.7907/egch-kp69

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 (µγ)γϵΓ on X such that µ = Σγµγ and ν = Σγγµγ. 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.

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