Von Neumann Algebras and Ergodic Theory

Vanderbilt University (Nashville, Tennessee)


Workshop Dates: April 27-28, 2013

Organizer: Jesse Peterson <jesse.d.peterson@vanderbilt.edu>


Schedule of Talks

All talks will take place in Stevenson Center 1432, and all coffee breaks will be down the hall in Stevenson Center 1425.

Saturday April 27th

8:30-9:00Coffee
9:00-9:50Lewis Bowen
10:00-10:30Coffee
10:30-11:20Robin Tucker-Drob
11:30-12:20Kostya Medynets
12:30-2:00Lunch
2:00-2:50Hanfeng Li
3:00-3:30Coffee
3:30-4:20Clinton Conley
4:30-5:20Ionut Chifan

Sunday April 28th

8:30-9:00Coffee
9:00-9:50Simon Thomas
10:00-10:30Coffee
10:30-11:20Artem Dudko
11:30-12:20Kate Juschenko
12:30-2:00Lunch
2:00-2:50Thomas Sinclair
3:00-3:50Darren Creutz

Program

Lewis Bowen

Cheeger Constants and L² Betti Numbers

The Cheeger constant of an infinite-volume manifold is the greatest lower bound of area to volume ratios of its compact sub-manifolds. It is related to the zero-th eigenvalue of the Laplace operator and, in the case of real hyperbolic manifolds, to the Hausdorff dimension of the limit set of the fundamental group. We ask: how small can the Cheeger constant be of a real hyperbolic n-manifold with no boundary and free fundamental group? I'll explain some partial results on this question by making use of L² Betti numbers.

Ionut Chifan

Structural Results for II₁ Factors of Negatively Curved Groups

In this talk I will discuss some recent results on the classification of von Neumann algebras arising from negatively curved groups and their measure preserving actions on probability spaces. This is based on a joint work with T. Sinclair and B. Udrea.

Clinton Conley

Local Complexity Among Treeable Equivalence Relations

Group-theoretic rigidity techniques such as Zimmer and Popa cocycle superrigidity have been instrumental in works of Adams-Kechris, Thomas, and Hjorth (among others) in realizing complexity in the partial order of Borel reducibility among countable Borel equivalence relations. We introduce an alternative elementary notion of rigidity which interacts favorably with Borel reducibility, allowing us to localize various complexity results to just above measure hyperfinite in the class of treeable equivalence relations. This is joint work with Ben Miller.

Darren Creutz

Stabilizers of Ergodic Actions of Product Groups and Lattices in Products

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of a semisimple real Lie group with every simple factor of higher-rank is essentially free and likewise for lattices in such groups. Bader and Shalom, following a different approach, showed that any properly ergodic action of a product of two simple groups with property (T) is essentially free, but their methods do not yield information about lattices.
I will present recent work expanding on the approach of Bader and Shalom generalizing the results of Stuck and Zimmer and of Bader and Shalom to the case when only one factor has (T) and obtaining a classification statement for actions of lattices in products of simple Howe-Moore groups.

Artem Dudko

On Characters of Approximately Finite Groups and the Higman-Thompson Groups

It is well-known that given a measure-preserving action of a group G on a measure-space (X,μ) the formula χ(g)=μ{x : gx=x} defines a character on G. Vershik conjectured that for a “rich” group G all characters can be obtained in this way. We verify Vershik’s conjecture for approximately finite groups (full groups of Bratteli diagrams) and the Higman-Thompson groups. The talk is based on joint papers with Konstantin Medynets.

Kate Juschenko

On Klondike of Amenable Groups

I will discuss amenable and recurrent actions of discrete groups. Sufficient conditions of amenability of a finitely generated group acting by homeomorphisms on a topological space will be presented. Surprisingly, this conditions are satisfied by large classes of groups: all known non elementary amenable groups, bounded and linearly growing automata groups. This is joint with V. Nekrashevych and M. de la Salle.

Hanfeng Li

Mean Dimension and von Neumann-Luck Rank

Mean dimension is a numerical invariant for continuous actions of countable amenable groups, coming up in topological dynamics and related to entropy. The von Neumann-Luck rank is an invariant for modules of the integral group ring of countable groups, coming up in L²-invariants and related to L²-Betti numbers. I will discuss the relation between the von Neumann-Luck rank of a module of the integral group ring of a countable amenable group and the mean dimension of the associated algebraic action. This is joint work with Bingbing Liang.

Kostya Medynets

Full Groups in Ergodic Theory

We will give an overview of the properties of full groups of ergodic measure-preserving equivalence relations. We will discuss the algebraic structure of full groups and its relation to the dynamical properties of the underlying equivalence relations.

Thomas Sinclair

Some Comments on Ergodic Theorems for Affine Actions on Hilbert Space

Simon Thomas

Invariant Random Subgroups of the Group of Finitary Permutations

I will discuss Vershik's classification of the invariant random subgroups of the group of finitary permutations of the natural numbers.

Robin Tucker-Drob

Mixing Actions of Countable Groups are Almost Free

A measure preserving action of a countably infinite group G is called totally ergodic if every infinite subgroup of G acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. I will discuss the proof that all totally ergodic actions are free modulo a finite kernel.
One consequence of this is a group theoretic characterization of countable groups whose non-trivial Bernoulli factors are all free. This characterization suggests an analogous one for groups whose non-trivial weak Bernoulli factors are all free. Whether this analogous characterization holds turns out to be closely related to open problems about reduced C*-algebras of countable groups.


Funding for the workshop is in part from the Shanks Foundation (Vanderbilt University).