posted on 2024-08-01, 00:00authored byAhmet Özkan Demir
We introduce and explore the notion of Measurable Imbedding, which gives a partial ordering among discrete countable groups. We show it behaves nicely under some group-theoretic operations like free products and graph products (generalizing the same results for Orbit Equivalence). Then, we explore rigidity and flexibility for this notion and prove some rigidity results for lattices in higher rank simple Lie groups (generalizing the same results for Measure Equivalence rigidity). Finally, we introduce a new invariant called measurable free rank, which can be thought of as a measurable version of noncommutative free subgroup rank of a group (introduced by Campagnolo and Kammeyer) and compute it for non-elementary hyperbolic groups as 1.
History
Advisor
Alex Furman
Department
Mathematics, Statistics, and Computer Science
Degree Grantor
University of Illinois Chicago
Degree Level
Doctoral
Degree name
Doctor of Philosophy
Committee Member
Daniel Groves
Osama Khalil
Wouter van Limbeek
Tsachik Gelander