Speakers:Université du Québec à Montréal
Title: Knot commensurability and the Berge conjecture
Abstract: In this talk I will report on joint work with Michel Boileau, Radu Cebanu and Geneviève Walsh. One of the main results is that the commensurability class of the complement of a hyperbolic knot without hidden symmetries contains at most two other knot complements. If it contains at least one other, then both are fibred of the same genus. We can completely characterize those periodic knots (i.e. there is an axis of symmetry disjoint from the knot) without hidden symmetries whose complement is commensurable with another hyperbolic knot complement. If it is not periodic, any such characterization involves a generalization of the Berge conjecture.
University of Utah
Title: Virtually Geometric Multiwords
Abstract: A multiword in a free group F is geometric if it can be realized as an embedded multicurve on the surface of a handlebody with fundamental group F. It is virtually geometric if it becomes geometric upon passing to a finite index subgroup. I will show that virtually geometric multiwords are those that are built from geometric pieces.
Title: Contact-graphs of CAT(0) cube complexes
Abstract: Associated to a CAT(0) cube complex is a pair of graphs, the ``crossing graph'' and the ``contact graph'', that keep track of the crossing-relation and the slightly more general ``contact'' relation on the set of hyperplanes. We have shown that for any CAT(0) cube complex, the contact-graph is quasi-isometric to a tree. We deduce from this fact that uniformly locally finite CAT(0) cube complexes embed in products of finitely many trees. We also obtain a new proof of Wright's recent result, that the asymptotic dimension of a CAT(0) cube complex is bounded above by its dimension.
Title: Mapping class groups of Heegaard splittings
Abstract: A Heegaard splitting is determined by a surface embedded in a 3-manifold in a certain nice way. The symmetries of this embedding determine a subgroup of the mapping class group of the surface, called the mapping class group of the Heegaard splitting. This group can also be described as the group of isometries of the curve complex preserving a certain subset, allowing one to study the group from both a geometric and a topological perspective. I will discuss recent progress on understanding mapping class groups of Heegaard splittings from these two perspectives.
University of Wisconsin
Title: Mapping class groups at different levels.
Abstract: It is a theorem of Bass, Lazard, and Serre, and, independently, Mennicke, that the special linear group SL(n,Z) enjoys the congruence subgroup property when n is at least 3. This property is most quickly described by saying that the profinite completion of the special linear group injects into the special linear group of the profinite completion of Z. There is a natural analog of this property for mapping class groups of surfaces. Namely, one may ask if the profinite completion of the mapping class group embeds in the outer automorphism group of the profinite completion of the surface group.
M. Boggi has a program to establish this property for mapping class groups. I'll discuss some partial results, and what remains to be done.
University of California, Berkeley
Title: Factorizing homomorphisms through extended Dehn fillings
Abstract: In this talk we introduce the notion of extended Dehn fillings. For a homomorphism $\phi:G\to H_\gamma$, where $G$ is a finitely presented group, and where $H_\gamma$ is the Dehn filling of a hyperbolic $3$-manifold group $H$ along a slope $\gamma$ on a torus cusp, we show that $\phi$ factorizes through the extended Dehn filling epimorphism if $\gamma$ is sufficiently near the cusp in the usual Jorgensen-Thurston sense. I will explain why `extended' cannot be removed from the statement here. This result is an essential ingredient in the recent proof of Simon's conjecture of Ian Agol and the speaker.
Title: Dynamics on character varieties: ergodicity, proper discontinuity and topology
Abstract: We discuss the dynamics of Out(F_n) on the PSL(2,C) character variety of F_n. Regions of proper discontinuity and ergodicity can be described, although a complete dynamical decomposition is still unknown. We will focus on a construction of representations associated to knot complements that indicates that the rank of a representation in the region of proper discontinuity need not be close to the rank of its image group, nor constrained by its geometry in any obvious way. Parts of this are joint work with T. Gelander and with Y. Moriah.
University of Illinois
Title: Dynamics of Irreducible Endomorphisms of the Free Group
Abstract: We extend recent dynamical techniques for studying automorphisms of the free group to injective endomorphisms. Focusing on the class of endomorphisms that satisfy the Bestvina-Handel irreducibility criterion, we characterize the dynamics on conjugacy classes and on the boundary of the free group via a "train track" theorem. After imposing a (necessary) additional hypothesis, we consider the action of an irreducible endomorphism on the closure of the Culler-Vogtmann Outer space and show that this action has "north dynamics"--there is a unique (attracting) fixed point in the interior, such that convergence to this point is uniform.
Title: On the Geometry of the Free Factorization Graph
Abstract: The group Out(F_n) of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out(F_n) acts, in analogy with the curve complex for the mapping class group. In this talk on joint research with Dima Savchuk, we'll discuss some results about the geometry of free factorization graph, a space on which Out(F_n) acts and a proposed candidate for a curve complex analogue. In particular, we have found arbitrary rank quasi-flats in the free splitting graph, showing it is not hyperbolic and has infinite asymptotic dimension. Thus, it is probably not the `correct' curve complex analogue.
Title: Dehn function and simple connectedness of the asymptotic cone of Lie groups.
Abstract: We prove that the Dehn function of a connected Lie group grows either polynomially or exponentially. Moreover, we characterize these two cases algebraically (in terms of weights in the Lie algebra). We also characterize Lie groups with simply connected asymptotic cone. Unexpectedly, we find a whole class of connected solvable Lie groups for which the Dehn function is at most cubic, but whose asymptotic cone is not simply connected. Some of these groups have lattices, and therefore provide polycyclic examples. We get similar results for algebraic groups over a local field, with an interesting and unexpected dependence on the field.