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Descriptive Set Theory and Dynamics of Countable Groups


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


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.

Item Type:Thesis (Dissertation (Ph.D.))
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.
Thesis Committee:
  • Tamuz, Omer (chair)
  • Conlon, David
  • Vidnyánszky, Zoltán
  • Kechris, Alexander S.
Defense Date:2 May 2022
Funding AgencyGrant Number
NSF Division of Mathematical Sciences1464475
NSF Division of Mathematical Sciences1950475
Record Number:CaltechTHESIS:05252022-040224796
Persistent URL:
Related URLs:
URLURL TypeDescription adapted for Chapter 4
Shinko, Forte0000-0001-8142-1509
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14608
Deposited By: Forte Shinko
Deposited On:27 May 2022 23:02
Last Modified:04 Aug 2022 23:36

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