CaltechTHESIS committee: Article
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 11 Nov 2024 07:28:18 -0800Analogues of Amenability
https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708
Year: 2021
DOI: 10.7907/yymw-xd55
In this thesis I study multiple notions related to and inspired by amenability coming from the points of the view of random walks on groups, dynamical systems, Borel equivalence relations, and descriptive linear algebra. In particular I study notions related to harmonic functions, invariant measures, hyperfiniteness, and dichotomy theorems.https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708Analogues of Amenability
https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708
Year: 2021
DOI: 10.7907/yymw-xd55
In this thesis I study multiple notions related to and inspired by amenability coming from the points of the view of random walks on groups, dynamical systems, Borel equivalence relations, and descriptive linear algebra. In particular I study notions related to harmonic functions, invariant measures, hyperfiniteness, and dichotomy theorems.https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708Descriptive Set Theory and Dynamics of Countable Groups
https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796
Year: 2022
DOI: 10.7907/egch-kp69
<p>This thesis comprises four papers.</p>
<p> 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.</p>
<p> 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. </p>
<p> 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<sub>γ</sub>)<sub>γϵΓ</sub> on X such that µ = Σ<sub>γ</sub>µ<sub>γ</sub> and ν = Σ<sub>γ</sub>γµ<sub>γ</sub>. We establish a generalization of this result to cardinal algebras.</p>
<p> 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.</p>https://resolver.caltech.edu/CaltechTHESIS:05252022-040224796