Highway CA-17
Groups, Geometry, and Topology Seminar
A joint seminar between San José State University and the University of California Santa Cruz. It meets once per term on Fridays.
March 8, 2024 (Friday), UC Santa Cruz, McHenry 4130
Speakers:
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.
Schedule:
9:30 - welcome
10:00-11:00 - Matt's Intro Talk
11:00-11:30 - coffee break
11:30-12:30 - Michelle's Intro Talk
12:30-2:00 - lunch break
2:00-3:00 - Matt's Research Talk
3:00-3:30 - coffee break
3:30-4:30 - Michelle's Research Talk
Past seminars:
November 3, 2023 (Friday), San José State University, MacQuarrie Hall 324
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.