Citation
Tsankov, Todor Dimitrov (2008) Amenability, Countable Equivalence Relations, and Their Full Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/N91CHV48. https://resolver.caltech.edu/CaltechETD:etd05022008113702
Abstract
This thesis consists of an introduction and four independent chapters.
In Chapter 1, we study homeomorphism groups of metrizable compactifications of the natural numbers. Those groups can be represented as almost zerodimensional Polishable subgroups of the group S_{∞}. We show that all Polish groups are continuous homomorphic images of almost zerodimensional Polishable subgroups of S_{∞}. We also find a sufficient condition for these groups to be one dimensional.
In Chapter 2, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding shift action of Γ on M^{X}, where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the corresponding shift has almost invariant sets. This is joint work with Alexander Kechris.
In Chapter 3, we prove that if the Koopman representation associated to a measurepreserving action of a countable group on a standard nonatomic probability space is nonamenable, then there does not exist a countabletoone Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an action on the boundary of a countably splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. This is joint work with Inessa Epstein.
In Chapter 4, we study full groups of countable, measurepreserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of generators of a dense subgroup of full groups allowing us to distinguish full groups of equivalence relations generated by free, ergodic actions of the free groups F_{n} and F_{m} if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group iff its full group has a finitely generated dense subgroup. This is joint work with John Kittrell.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  almost invariant sets; almost zerodimensional; amenable actions; amenable representations; compactifications; full groups; generalized Bernoulli shifts; modular actions; orbit equivalence; Polishable ideals; strong ergodicity 
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, 2007. Scott Russell Johnson Prize Graduate Dissertation Prize in Mathematics, 2007. 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  29 April 2008 
Record Number:  CaltechETD:etd05022008113702 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd05022008113702 
DOI:  10.7907/N91CHV48 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  1584 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  20 May 2008 
Last Modified:  03 Dec 2019 22:30 
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