A joint seminar between San José State University and the University of California Santa Cruz. It meets once per term on Fridays.
Please register by February 21 at this link if you would like lunch.
Becky Eastham, University of California - Riverside
Intro Talk: A brief intro to combinatorial group theory and Out( Fₙ )
We will talk about some prerequisite material and motivation for defining and working with the Whitehead complex. In particular, we'll talk about the analogy between mapping class groups of surfaces and Out( Fₙ ) and the spaces on which each group acts. We will also review/introduce some concepts from combinatorial group theory, including Van Kampen diagrams, which play an essential role in the proof that the Whitehead complex of the rose is nonhyperbolic.
Seminar Talk: Separable homology of graphs and the Whitehead complex
We will introduce the Whitehead complex, which is a 1-complex associated with a finite cover of the rose. We show that it is connected if and only if the fundamental group of the associated cover is generated by elements in a proper free factor of the free group. When the associated cover represents a characteristic subgroup of the free group, the complex admits an action of Out( Fₙ ) by isometries. We then explore the coarse geometry of the 1-complex, showing that every component has infinite diameter, and that the 1-complex associated with the rose is nonhyperbolic. As corollaries, we obtain that the Cayley graph of the free group with the infinite generating set consisting of all primitive elements is nonhyperbolic. Time permitting, we will discuss future work related to the Whitehead complex.
Paige Hillen, University of California - Santa Barbara
Intro Talk: Dynamics of Out( Fᵣ ) and Outer Space
Given an outer automorphism f of the free group Fᵣ and a word w in Fᵣ , how fast does the cyclically reduced word length of fⁿ (w) grow as n increases? When f is irreducible, fⁿ (w) has the same exponential asymptotic growth rate for every word w, called the stretch factor. We’ll talk about how to use graph maps to compute this stretch factor, and what this value tells us about the dynamics of f. In particular, Out( Fᵣ ) acts on Outer Space, a space of marked metric rank r graphs, and the stretch factor of f gives information about the action of f on Outer Space.
Research Talk: Minimal Stretch Factors and Graph Symmetry
Stretch factors are always a type of algebraic integer called a weak Perron number. Conversely, Thurston showed that every weak Perron number is the stretch factor of some outer automorphism of Fᵣ . However, given a weak Perron number, there is no control on how large the rank r of the corresponding free group must be. In particular, the exact set of values that can occur as stretch factors for a fixed rank r is unknown. We’ll discuss recent work finding the minimal stretch factor among irreducible outer automorphisms of F₃ . The framework for finding this minimum relies on understanding the symmetry of graphs which admit optimal graph map representatives of outer automorphisms.
Schedule:
9:30 - welcome
10:00-11:00 - Intro Talk 1
11:00-11:30 - coffee break
11:30-12:30 - Intro Talk 2
12:30-2:00 - lunch break
2:00-3:00 - Research Talk 1
3:00-3:30 - coffee break
3:30-4:30 - Research Talk 2
Gil Goffer UC San Diego
Intro Talk: Small Cancellation Theory
Small cancellation theory studies groups given by group presentations in which the defining relations have “small overlaps” with each other. In the talk I’ll explain the ‘basic rules’ of small cancellation, and how to use them to produce groups with desired properties.
Research Talk: Can group laws be learned using random walks?
In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability larger than 5/8, must be abelian. In the talk I’ll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Greenfeld and Olshanskii, answering questions by Amir, Blachar, Gerasimova, and Kozma.
Alex Rasmussen Stanford
Intro Talk: Dimension, curvature, and surfaces
Curvature and dimension are intuitive notions from our visual experience. It turns out that notions of curvature and dimension can be defined in great generality. In this talk we will define hyperbolicity and asymptotic dimension for metric spaces and study some examples. We will then define the curve graph of a surface and see how these notions relate to curve graphs.
Research Talk: Disintegrating curve graphs
Curve graphs are crucial tools for studying mapping class groups of surfaces. However, many basic questions on their geometry remain open. In this talk, we will shed light on the geometry of curve graphs by describing “filtrations” of them by hyperbolic graphs. These filtrations yield quasi-isometric disintegrations of curve graphs into trees. As a corollary, we provide a new proof of finite asymptotic dimension of curve graphs. Finally, we describe some useful aspects of the dynamics of the mapping class group actions on the graphs in the filtrations.
Michelle Chu (University of Minnesota - Twin Cities): The character variety and surfaces in 3-manifolds
Intro Talk: Given a surface S in a 3-manifold M, there is an associated tree T and an action of the fundamental group of M on T which determines the surface S. In the reverse direction, given a nice action of a 3-manifold group on a tree, there is an associated surface on the 3-manifold determined by the action on the tree. I will discuss these constructions and introduce how the character variety can be used to find such actions of 3-manifold groups on trees.
Research Talk:
The SL(2,C)-character variety of the fundamental group of a 3-manifold encodes a lot of topological and geometric information about the manifold. In this talk I will discuss both old and new results in the case of two-bridge knot complements.
Matthew Durham (University of California - Riverside): An asymptotically CAT(0) structure on the mapping class group via cubical approximations
Intro Talk:
In this introductory talk, I will discuss and motivate the use of coarse notions of curvature for studying infinite groups and how they allow us to define boundaries for certain classes of groups. The first part of the talk will focus on CAT(0) groups and their visual boundaries. In the second part, I will discuss the notion of an asymptotically CAT(0) group, which was introduced by Aditi Kar and generalizes these more commonly studied settings. Many techniques generalize nicely to this setting and I will discuss a new visual boundary construction for these groups.
Research Talk:
In this talk, I will discuss work in preparation with Yair Minsky and Alessandro Sisto in which we prove that mapping class groups of surfaces are asymptotically CAT(0). In the first half of the talk, I will focus on some motivating applications, including building a Z-boundary for mapping class groups in the sense of Bestvina, and proving the Farrell-Jones conjecture for them, recovering a theorem of Bartels-Bestvina. In the second half, I will give some idea of how we build an asymptotically CAT(0) metric for the mapping class group using local approximations by CAT(0) cube complexes.
Anna Parlak (University of California Davis)
Intro Talk: Introduction to pseudo-Anosov flows
The aim of the talk is to present a topologist's perspective on pseudo-Anosov flows on 3-manifolds. I will start by analyzing topological properties of the geodesic flow on the unit tangent bundle of the hyperbolic plane. This will be used to define a larger class of flows on 3-manifolds, called Anosov flows, and their further generalization: pseudo-Anosov flows. I will discuss a few basic techniques of producing pseudo-Anosov flows and mention some connections between pseudo-Anosov flows and other aspects of 3-manifold topology/geometry/group theory.
Research Talk: Pseudo-Anosov flows and the Thurston norm
A pseudo-Anosov flow on a closed 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is spanned by homology classes of surfaces almost transverse to the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparameterization, flow which dynamically represents the face. Mosher found sufficient conditions on a non-circular flow to dynamically represent a non-fibered face, but the problem of the existence and uniqueness of the flow for every non-fibered face was unresolved. I will outline the proof of the fact that a non-fibered face can be in fact dynamically represented by multiple topologically inequivalent flows, and discuss how two distinct flows representing the same face may be related.
Mark Pengitore (University of Virginia)
Intro Talk: The word problem for finitely generated groups and residual finiteness
This talk will give an introduction to the study of the word problem using residual finiteness for finitely generated groups.
Research Talk: Residual finiteness growth functions of surface groups with respect to characteristic quotients
Residual finiteness growth functions of groups have attracted much interest in recent years. These are functions that roughly measure the complexity of the finite quotients needed to separate particular group elements from the identity in terms of word length. In this talk, we study the growth rate of these functions adapted to finite characteristic quotients. One potential application of this result is towards linearity of the mapping class group.