Minicourse abstracts
Speaker: Matthew Durham (University of California, Riverside)
Title: Semihyperbolicity of the mapping class group via stable cubulations
Abstract: In this minicourse, I will present a streamlined version of my work with Minsky and Sisto, in which we proved that the mapping class group is semihyperbolic, i.e. admits an equivariant quasi-geodesic bicombing. Our proof utilizes the fact, due to Behrstock-Hagen-Sisto, that the hierarchical hull of a finite set of points in the mapping class group is quasi-isometrically modeled by a cube complex. I will outline a new proof of this fact and also a simplified exposition of how we stablize their construction. The simplification comes from focusing on the case of modeling the hull of a pair of points, which is sufficient for building the family of paths which constitute the bicombing.
Introductory speaker: Jacob Garcia (University of California, Riverside)
Speaker: Anne Lonjou (Université Paris-Saclay)
Title: Cremona groups and CAT(0) cube complexes
Abstract: In this mini-course we will focus on the group of birational transformations of the projective plane called the Cremona group. A birational transformation is an isomorphism between two open dense subsets. This group is central in birational geometry. Even though this group is coming from algrebraic geometry, using geometric group theory tools have been a stepping-stone into its understanding. In this mini-course, I will introduce the blow-up cube complex, a CAT(0) cube complexe that we constructed with C. Urech. After explaining why this cube complex is CAT(0), we will see that there exists a really nice dictionary between some algebraic properties of this group and some geometric properties of the action on this complex. This will lead us to the following question, partially answered in a common work with A. Genevois and C. Urech. Consider a finitely generated subgroup G of the Cremona group such that any element induces an elliptic isometry (for this action): Does G admit a global fixed point?
Introductory speaker: Kejia Zhu (University of Illinois at Chicago)
Speaker: Marissa Loving (University of Wisconsin-Madison)
Title: Infinite-type surfaces, end-periodic homeomorphisms, and the geometry of 3-manifolds
Abstract: I will present work with Elizabeth Field, Heejoung Kim, Autumn Kent, and Chris Leininger (in various configurations) to understand the geometry of certain compact 3-manifolds with boundary which are obtained from the mapping tori of strongly irreducible end-periodic homeomorphisms. Leveraging subsurface projection and the coarse geometry of the curve complex, we use the dynamics of irreducible end-periodic homeomorphisms to produce an asymptotically sharp upper bound on the volumes of these manifolds (see arxiv:2106.15642). In ongoing work, we adapt the powerful machinery of pleated surfaces to the infinite-type setting to produce a corresponding lower bound on volume. In this mini-course, I will sketch the proofs of both the upper and lower bounds, with the goal of detailing our model manifold construction, and introducing our (new) key tool of well-pleated surfaces.
Introductory speaker: Brandis Whitfield (Temple University)
Speaker: Jean-Pierre Mutanguha (Institute for Advanced Study)
Title: The dynamics and geometry of free group endomorphisms
Abstract: I will start with a brief discussion of train track maps (and their various enhancements) for free group automorphisms. These ideas do not immediately generalize to nonsurjective free group endomorphisms. Using Stallings graphs, I give an alternative and elementary approach for studying endomorphisms. This is well-suited for reducing questions about free group endomorphisms to the automorphism case. I'll then sketch two applications: finding the fixed point subgroup of an endomorphism; and characterizing when an endomorphism's mapping torus is hyperbolic.
Introductory speaker: Kailey Perry (University of Arkansas)