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Topology Festival

May 6–9, 2011

Abstracts of Talks

Matthew Foreman, University of California at Irvine

Classifying Measure Preserving Diffeomorphism of the Torus

Relatively recent joint work with Rudolph and Weiss showed rigorously that it is impossible to classify ergodic measure preserving transformations of the unit interval. In joint work with Weiss in 2010, this result was extended to diffeomorphisms of the torus. I will survey these results and explain a very recent result that places the isomorphism relation for diffeomorphisms above the “Graph Isomorphism Problem” and hence shows that it is complete for S actions.

Rostislav Grigorchuk, Texas A&M University

Manifestations of the Lamplighter

A group belongs to The Lamplighter if it is a wreath product of a finite group and an infinite group. The most classical examples are the wreath products of Z2 by Zn, but one can consider more sophisticated examples involving free groups, groups of intermediate growth or Burnside groups as the active group. We will present a survey of results from algebra, topology, amenability, growth, random walks, and operator algebras which were obtained using the Lamplighter construction.

Grigorchuk Lecture Notes (PDF)

Olga Kharlampovich, McGill University

Algebraic Geometry for Groups

In this talk I will discuss some new research areas, methods, and results which appeared in group theory in connection to solutions of Tarski problems about first order theory of free groups. I will also discuss elemenary classification questions and f.g. groups universally equivalent to free groups and elementary equivalent to free groups.

New areas: Algebraic geometry over groups, limit groups, groups acting freely on non-Archimedean trees, free actions on Λ-hyperbolic spaces, algebraic theory of equations in groups.

New methods: Elimination processes (dynamical processes, transformations similar to interval exchange), regular actions, non-Archimedean words and presentations, Lyndon’s completions.

Kharlampovich Workshop Notes (PDF)

Kharlampovich Lecture Notes (PDF)

Darren Long, University of California at Santa Barbara

Some Algebraic Applications of Real Projective Manifolds

We use ideas from real projective manifolds to give some new representations of surface groups. Some structural aspects may also be discussed.

Jason Manning, State University of New York at Buffalo

Hyperbolic Dehn Filling of Spaces and Groups

Dehn filling is a classical tool in 3-dimensional topology, in which a 3-manifold with torus boundary is “filled” to obtain a closed 3-manifold. Thurston showed that if the bounded 3-manifold has interior with a hyperbolic metric, then so do most fillings. I’ll talk about what it means to “fill” higher dimensional cusped hyperbolic manifolds, and how to generalize Thurston’s result to this setting. I’ll also talk about what it means to “fill” a relatively hyperbolic group and applications of this idea. I hope to give an overview of joint work between myself and Agol, Groves, Fujiwara, and Martinez-Pedroza, as well as related work by Osin and others.

Yair Minsky, Yale University

Dynamics of Automorphism Groups: Ergodicity, Stability, and Topology

The representation space Hom (Γ, G) admits a natural action by automorphisms of Γ. When G is the group of isometries of hyperbolic space we can compare the dynamical structure of this action with the geometry (e.g., discreteness) of the representations themselves. Our understanding is incomplete but, when Γ is a free group, we can locate regions where the action is properly discontinuous, and regions where it is ergodic (but non-mixing). We can also construct examples associated to presentations of knot groups which shed some (but not much) light on the relation between dynamics of automorphisms and topology of quotient 3-manifolds. Some of this is joint work with T. Gelander and with Y. Moriah.

Justin Moore, Cornell University

Amenability and Ramsey Theory

In 2005, Kechris, Pestov and Todorcevic proved a result which essentially equates the study of the extreme amenablity of automorphism groups of countable structures with structural Ramsey theory. Here extreme amenability is the assertion that every continuous action of the group on a compact space has a fixed point. At the time it was unclear whether amenability has an analogous connection to Ramsey theory. I will show that it does. This involves the isolation of a considerable weakening of the Følner criteria for a group. It also has the following consequence: if G is non amenable group, then there is a subset E of G such that no finitely additive probability measure on G measures all translates of E equally. I will also discuss these results in the context of the (still open) amenability problem for Thompson’s group F.

Alexandra Pettet, Oxford University

Abstract Commensurators of the Johnson Filtration

The Torelli group is the subgroup of the mapping class group which acts trivially on the homology of the surface. It is the first term of the Johnson filtration, the sequence of subgroups which act trivially on the surface group modulo some term of its lower central series. We prove that the abstract commensurator of each of these subgroups is the extended mapping class group. This is joint work with Martin Bridson and Juan Souto.

Slawek Solecki, University of Illinois at Urbana-Champaign

Fixed Points, Ramsey Theorems, Concentration of Measure, and Submeasures

I will survey recently discovered connections between fixed point theorems for actions of large groups and the Fraïssé limit construction from model theory, Ramsey theorems, concentration of measure, and “geometry of submeasures.” Groups that will be of interest to us are closed subgroups of the group of all permutations of natural numbers and groups of the form L0(φ, K), where φ is a submeasure and K is a locally compact abelian group. The Ramsey theorems that are relevant to our considerations are Ramsey theorems for finite structures and more “exotic” Ramsey theorems with roots in algebraic topology.

Solecki Lecture Notes (PDF)

Simon Thomas, Rutgers University (2 talks)

A Descriptive View of Geometric Group Theory (Workshop)

Gromov’s geometric group theory seeks to classify finitely generated groups in terms of the “large scale” geometry of their Cayley graphs. At first glance, this program appears to be even more difficult than that of classifying finitely generated groups up to isomorphism. In this talk, after introducing some of the basic notions of geometric group theory and descriptive set theory, I will consider the question of whether this is indeed the case.

Thomas Workshop Notes (PDF)

The Complexity of the Quasi-Isometry Relation for Finitely Generated Groups

In this talk, I will explain what is currently known concerning the complexity of the quasi-isometry relation for finitely generated groups and discuss a number of open problems.

Thomas Lecture Notes (PDF)