## Logic Seminar

Patrick Lutz

UC Berkeley

Lossless expansion and measure hyperfiniteness

**Abstract:**The study of the structure of countable Borel equivalence relations under Borel reducibility has been a major focus of descriptive set theory over the past few decades. However, many open questions remain, many of which involve the hyperfinite equivalence relations (essentially the simplest nontrivial countable Borel equivalence relations). In order to better understand these questions, Conley and Miller introduced a weakening of Borel reducibility, known as measure reducibility. They then answered the analogues for measure reducibility of several of these questions. However, they left open at least one such question. Namely, is there a minimal (in the sense of measure reducibility) non-hyperfinite equivalence relation? Such an object is called a "measure successor of E_0." In ongoing work, Jan Grebik and I have isolated a combinatorial property of group actions on Polish spaces which implies that the associated orbit equivalence relation is a measure successor of E_0 and found several examples of group actions which are plausible candidates for satisfying this condition. The combinatorial property we have identified is a strong form of expansion which we call "lossless expansion" after a similar property studied in computer science and combinatorics. I will explain the context for Conley and Miller's question and the combinatorial condition that Grebik and I have isolated.

Wednesday April 17, 2024 at 4:00 PM in 712 SEO