Redbud Topology
Conference
April 11-13, 2014
University of Oklahoma
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The dynamical behavior of a homeomorphism of a surface was classified by Nielsen in the 1920s-1940s. His work was overlooked until the 1970s when Thurston, unaware of Nielsen's papers, rediscovered this classification. Since this time, the Nielsen-Thurston classification of surface homeomorphisms has become an indispensable tool in the theory of surfaces and three dimensional manifolds.
Bestvina and Handel, among others, applied Thurston's approach to the study of automorphisms of free groups. As a result, these two theories have developed in parallel, oftentimes with insights from surfaces leading to insights in free groups.
I will describe the dynamical picture in both settings.
A finitely generated group that splits as a (semidirect product) free-by-cyclic group can often be expressed as such in infinitely many ways. This talk will explore the different ways a given group G splits as a free-by-cyclic group or, more generally, as an ascending HNN-extension over a finitely generated free group. It turns out that this algebraic question is closely related the the dynamical problem of finding "cross sections" of a semiflow on a certain 2-complex. After explaining this correspondence I will introduce a polynomial invariant that exactly calculates the set of cross sections and thereby identifies an interrelated family splittings of G. In fact, this family corresponds to a full connected component of the BNS-invariant of G. This is joint work with Ilya Kapovich and Christopher Leininger.
Most problems about finitely presented groups are algorithmically unsolvable in general. However, if one makes extra hypotheses about the group, then often a lot can be said. In this work, we assume there is a solution to the word problem.
The main theorem is that given a solution to the word problem one can decide whether or not a finite presentation defines a group that is the fundamental group of a closed 3-manifold.
(This is joint work with Henry Wilton and Jason Manning.)
In this talk I will describe some general combinatorial results that relate to the study of BNS invariants for finitely generated groups. These include classical properties of Perron-Frobenius digraphs and their characteristic polynomials, McMullen's work on Teichmueller polynomials and his more recent work on metric digraphs, and my joint work with Algom-Kfir and Rafi on dual digraphs and cycle complexes associated to the monodromy of free-by-cyclic groups.
In this talk I will describe work of Thurston, Fried, and McMullen on 3-manifolds that fiber over the circle. Their analysis provides a wealth of information about the relationships between different fibrations and the associated monodromies. Furthermore, by work of Stallings, Bieri-Neumann-Strebel, and others, one can interpret much of this information in purely algebraic terms, providing impetus for generalizations to other contexts.
The geometric intersection number for a pair of curves living on a surface is a fundamental notion whose utility for understanding mapping class groups is hard to overstate. Guirardel, building on work of Scott and others, generalized this notion to a pair of trees acted on by a group. I will discuss some ways of seeing this intersection number and some mapping-class-inspired applications for free group automorphisms. In particular, a simple algorithm for determining Nielsen-Thurston classification of a mapping class (joint with Thomas Koberda) has an analogue for deciding whether a free group outer automorphism is fully irreducible (joint with Matt Clay and Alexandra Pettet); both depend on being able to estimate intersection number between curves / trees and their images under a mapping class / free group automorphism.
In this talk I'll go into some of the background material relevant for my talk on Saturday. Specifically, I'll talk about Sageev's construction of cube complexes from codimension one subgroups, the Haglund-Wise notion of *special* cube complexes, and various characterizations of these cube complexes in the hyperbolic setting.
Wise's malnormal special quotient theorem (MSQT) is a key ingredient in the recent resolution by Agol of some central conjectures in 3-manifolds, including the virtual Haken conjecture. The MSQT allows one to perform certain small-cancellation operations or "Dehn fillings" on a virtually special hyperbolic group, in such a way that the result is still hyperbolic and virtually special. I'll give an idea how this theorem is used and outline a new proof by Agol, Groves, and myself.
I will discuss a vanishing result for certain homology groups of infinite volume, rank one, locally symmetric manifolds. The talk will focus on connections between geometric and topological complexity. This work is joint with Chris Connell and Benson Farb.
The Masur criterion for the mapping class group says that the vertical foliation of a recurrent Teichmueller geodesic is uniquely ergodic. We discuss how this criterion can be adapted to the outer automorphism group of a free group acting on outer space, and sketch some applications. This is a joint project with Hossein Namazi and Patrick Reynolds.
There are many analogies between the outer automorphism group of a free group Out(F) and the mapping class group of a surface Mod(S). I'll explain how each of these groups contains many right-angled Artin subgroups and how these subgroups can be used to understand the structure of both Mod(S) and Out(F). Interestingly, attempting to understand the properties of elements in right-angled Artin subgroups also reveals some major differences between Out(F) and Mod(S). I'll explain these differences and how they affect the study of Out(F).
Though quasimorphisms on a free group (serving as a model for hyperbolic groups) are plentiful, it is often difficult to explicitly construct a quasimorphism that one knows must exist. I'll describe some results in this area (joint with Danny Calegari) and some surprisingly basic open questions.