## 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_{γ}µ_{γ} and ν = Σ_{γ}γµ_{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.

Item Type: | Thesis (Dissertation (Ph.D.)) | ||||||
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Subject Keywords: | descriptive set theory, logic, borel, group, amenable | ||||||

Degree Grantor: | California Institute of Technology | ||||||

Division: | Physics, Mathematics and Astronomy | ||||||

Major Option: | Mathematics | ||||||

Awards: | Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2022. Apostol Award for Excellence in Teaching in Mathematics, 2020, 2021, 2022. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2021. | ||||||

Thesis Availability: | Public (worldwide access) | ||||||

Research Advisor(s): | - Kechris, Alexander S.
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Thesis Committee: | - Tamuz, Omer (chair)
- Conlon, David
- Vidnyánszky, Zoltán
- Kechris, Alexander S.
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Defense Date: | 2 May 2022 | ||||||

Funders: |
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Record Number: | CaltechTHESIS:05252022-040224796 | ||||||

Persistent URL: | https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796 | ||||||

DOI: | 10.7907/egch-kp69 | ||||||

Related URLs: |
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ORCID: |
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Default Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||

ID Code: | 14608 | ||||||

Collection: | CaltechTHESIS | ||||||

Deposited By: | Forte Shinko | ||||||

Deposited On: | 27 May 2022 23:02 | ||||||

Last Modified: | 04 Aug 2022 23:36 |

## Thesis Files

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