### Saturday, Oct 24, 2015

#### 9:30 a.m. – 6 p.m. with coffee and lunch breaks

### Carnegie Mellon University

# Su Gao : “Countable abelian group actions”

## Description

A number of “stubbornly open” problems about countable Borel equivalence relations concern hyperfiniteness. For instance, the increasing union problem asks if the increasing union of a sequence of hyperfinite equivalence relations is still hyperfinite. In the past decade or so, the only progress on hyperfiniteness problems has been the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions (Gao and Jackson, “Countable abelian group actions and hyperfinite equivalence relations”, Inventiones Mathematicae, 2015) and then the extension of this result to locally nilpotent groups (Schneider and Seward, “Locally nilpotent groups and hyperfinite equivalence relations”, to appear).

The hyperfiniteness proofs are based on an elaborate theory of Borel marker structures with regularity properties. Now researchers have a good understanding of which regularity properties are possible and which are beyond hope. For the proofs of negative results two new concepts and methods have been playing a key role. One of them is the introduction and construction of hyperaperiodic elements with various additional properties. The other is the introduction of new forcing notions that are special cases of the so-called orbit forcing. The workshop will be roughly divided into four lectures:

- In the first lecture we will construct some basic regular marker structures for the Bernoulli shift of ℤ
^{n}. Using such marker structures, we will give an outline of the proof of hyperfiniteness for orbit equivalence relations of countable abelian group actions. - In the second lecture we will give some advanced constructions of regular marker structures and use them to illustrate a proof that the Borel chromatic number for the free part of 2
^{ℤn}is 3. - In the third lecture we will construct some hyperaperiodic elements for 2
^{ℤn}and use them to show that the continuous chromatic number for the free part of 2^{ℤn}is 4. - In the fourth lecture we will consider some forcing constructions and use them to show that certain regular Borel marker structures do not exist.