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Descriptive Set Theory and the Ergodic Theory of Countable Groups


Tucker-Drob, Robin Daniel (2013) Descriptive Set Theory and the Ergodic Theory of Countable Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ER96-5D06.


The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel reducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of residually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abért and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirming a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how approximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group's reduced C*-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:descriptive set theory; ergodic theory
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2012. Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2013.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kechris, Alexander S.
Thesis Committee:
  • Kechris, Alexander S. (chair)
  • Makarov, Nikolai G.
  • Ramakrishnan, Dinakar
  • Sokic, Miodrag
Defense Date:17 April 2013
Funding AgencyGrant Number
National Science FoundationDMS-0968710
Record Number:CaltechTHESIS:05162013-102038765
Persistent URL:
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Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7716
Deposited By: Robin Tucker-Drob
Deposited On:22 May 2013 22:22
Last Modified:04 Oct 2019 00:00

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