McGill ergodic group theory day

Dec. 7th, 2011

McGill ergodic group theory day

Dec. 7th, 2011

Matthew de Courcy-Ireland

Title: Tarski's Theorem: Amenability vs Paradox

Abstract: We prove Tarski's Theorem that if a group G acts on a set X, then there is a finitely additive G-invariant measure on X assigning measure 1 to a subset E of X if and only if E is not G-paradoxical. In particular, with G acting on X = E = G by left multiplication, Tarski's Theorem implies that a group if amenable if and only if its regular action on itself is not paradoxical. Time permitting, we discuss some applications of Tarski's beautiful result and give some definitions that will be needed for some of the day's other talks.

Mark Hagen

Title: L2 Betti numbers, nonpositive immersions, and the energy criterion

Abstract: After providing brief background on group von Neumann algebras and L2 Betti numbers, we describe the energy criterion, due to Wise, for the vanishing of the second L2 Betti number b_2(X,G) for a cocompact free G-complex X. We discuss some examples and applications of the energy criterion to the `nonpositive immersions property` and hence to local indicability.

Schedule:

14:30-15:00 Maxime Bergeron

16:30-17:00 Mark Hagen

16:00-16:30 Atefeh Mohajeri

15:00-15:30 Hadi Bigdely

14:00-14:30 Matthew de Courcy-Ireland

17:00-17:30 Keivan Mallahi Karai

Coffee Break - discussions

Abstracts:

Invariant Measures, Expanders and Property T

Rank Gradient, Cost of Groups and an application towards Rank Versus Heegaard Genus Problem

Tarski's Theorem: Amenability versus Paradox

L2 Betti numbers, nonpositive immersions, and the energy criterion

Product replacement algorithm and property T

Hjorth's lemma for treeable equivalence relations on standard Borel probability spaces

Atefeh Mohajeri

Title: Hjorth's lemma for treeable equivalence relations on standard Borel probability spaces

Abstract: Hjorth proved that any treeable equivalence relation on a standard Borel probability space with countable classes arises from an essentially free measure preserving action of a countable group G. Moreover, the relation is of integer cost if and only if G can be chosen to be free. We will discuss the proof of this result in detail.

Location: Talks start at 2pm in Burnside Hall, room 920.

Maxime Bergeron

Title: Invariant Measures, Expanders and Property T

Abstract: Banach and Ruziewicz asked whether the Lebesgue measure is the only finitely additive rotation invariant measure defined on the Lebesgue subsets of the sphere. Working towards an answer to this question, we present an arithmetic proof of the existence of free groups in some special orthogonal groups using the Hurwitz quaternions. This yields a paradoxical decomposition as in the Banach-Tarksi paradox. The problem is then related to various group theoretic notions such as amenability and Kazhdan's property T leading to some intriguing connections with expanders.

Hadi Bigdely

Title: Rank Gradient, Cost of Groups and an application towards Rank Versus Heegaard Genus Problem

Abstract: M. Abert and N. Nikolov provided a relation between the rank gradient of a Farber chain and the cost of the action of the group on the boundary of the coset tree. As an application, this shows the incompatibility of two well-known problems, Rank vs Heegaard genus conjecture and Fixed Price problem.

Rank vs Heegaard genus conjecture states that the Heegaard genus of any compact, orientable, hyperbolic 3-manifold equals the rank of its fundamental group. And Fixed Price problem asks if every countable group have fixed price. In this talk, after providing some backgrounds, we will describe the proof of the results.

Keivan Karai

Title: Product replacement algorithm and property T

Abstract: I will discuss the product replacement algorithm and the work of Lubotzky-Pak relating it to Property (T). If time allows, I will show how Lubotzky-Pak theorem can be quantified.