Chris Leininger (Rice University)
Title: Surface bundles and their coarse geometry (Parts I & II)
Abstract
I’ll discuss surface bundles over various spaces with a focus on their monodromy representations and how this influences properties of the fundamental group. In the first talk, I will explain Farb and Mosher’s notion of convex cocompactness in the mapping class group, its various incarnations, and its connections to the coarse hyperbolicity of surface bundles. Then I’ll describe more recent work generalizing convex cocompactness to notions of geometric finiteness, examples that illustrate hyperbolicity features, and mention several open problems. In the second talk, I’ll present a new construction of surface bundles over surfaces, providing the first examples of such bundles that are atoroidal.
The surface bundle construction in the second talk represents recent joint work with Autumn Kent. The work on convex cocompactness is also joint with Kent, as well as with Bestvina, Bromberg, Dowdall, Russell, and Schleimer in various combinations. The work on geometric finiteness is joint with Dowdall, Durham, and Sisto.
Hyungryul Baik (KAIST)
Title: Bifoliations on the plane and flows in 3-manifolds
Abstract
The orbit space of a (pseudo-)Anosov flows in 3-manifold is an example of bifoliated plane. Barthelmé-Bonatti-Mann recently gave a necessary and sufficient condition for a pair of circle prelaminations to be induced from a so-called pA-bifoliated plane (the class of pA-bifoliated planes includes all bifoliated planes from (pseudo-)Anosov flows in 3-manifolds). We first discuss this correspondence and then discuss how to reconstruct the 3-manifold and its flow from a group action on the bifoliated plane. This talk is based on the joint work with Chenxi Wu and Bojun Zhao.
Inhyeok Choi (Cornell University / KIAS)
Title: Genericity of pseudo-Anosovs and quasi-isometries
Abstract
In this talk, I will explain a recent result that pseudo-Anosov mapping classes are generic in every Cayley graph of mapping class groups. If time permits, I will also explain why this strategy goes well with quasi-isometries and implies genericity of Morse elements for groups quasi-isometric to (many) 3-manifold groups and special cubical groups.
David Fisher (Rice University)
Title: Finiteness of totally geodesic submanifolds
Abstract
Let (M, g) be a compact, negatively curved analytic Riemannian manifold. If M admits infinitely many closed immersed totally geodesic hypersurfaces than M has constant curvature and is a hyperbolic manifold. This can be read as the converse statement that all negatively curved analytic Riemannian manifolds that are not hyperbolic have only finitely many closed, immersed totally geodesic submanifolds. I will explain some motivations and the key ideas in the proof.
Koji Fujiwara (Kyoto University)
Title: The Hardy-Littlewood maximal inequality for hyperbolic groups
Abstract
The Hardy–Littlewood maximal inequality is a fundamental inequality for a maximal operator for an L¹ functions on Euclidean spaces. Naor–Tao gave an interesting geometric proof of the Hardy–Littlewood maximal inequality for a free group. I will explain that their argument applies to hyperbolic groups. This is a joint work with Amos Nevo.
Inkang Kim (KIAS)
Title: Free group actions in various geometric structures
Abstract
The free group with n generators Fₙ acts on various geometric structures such as hyperbolic geometry, real projective geometry and Hermitian symmetric spaces. We want to study the moduli space and characterize such actions using some invariants.
Sang-hyun Kim (KIAS)
Title: First order rigidity of manifold homeomorphism groups
Abstract
Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker’s theorem on isomorphic homeomorphism groups (1963) without relying on it. We also establish the analogous result for volume-preserving subgroups. Joint work with Thomas Koberda (Virginia) and Javier de la Nuez-Gonzalez (KIAS).
Sanghoon Kwak (KIAS)
Title: Nonunique Ergodicity on the Boundary of Outer Space
Abstract
The Culler–Vogtmann’s Outer space CVₙ is a space of marked metric graphs, and it compactifies to a set of Fₙ-trees. Each Fn-tree on the boundary of Outer space is equipped with a length measure, and varying length measures on a topological Fₙ-tree gives a simplex in the boundary. The extremal points of the simplex correspond to ergodic length measures. By the results of Gabai and Lenzhen-Masur, the maximal simplex of transverse measures on a fixed filling geodesic lamination on a complete hyperbolic surface of genus g has dimension 3g − 4. In this talk, we give the maximal simplex of length measures on an arational Fₙ-tree has dimension in the interval [2n − 7, 2n − 2]. This is a joint work with Mladen Bestvina, and Elizabeth Field.
Khánh Lê (Rice University)
Title: Order-preserving outer automorphisms of free and surface groups
Abstract
A group is bi-orderable if it admits a total ordering that is left and right invariant. Orderable groups have received recent attention due to their connection with dynamical group theory and with 3-manifold groups via the L-space conjecture. Given a bi- orderable group G, it is natural to ask which outer automorphisms of G preserve a bi-ordering on G since these correspond precisely to cyclic extension of G that is bi-orderable. Motivated by the connection with 3-manifolds, we focus on the case when G is a free group or a bi-orderable surface group. In this talk, I will give a complete characterization of order-preserving finite subgroups of outer automorphisms of non-abelian free and surface groups. I will also describe a criterion for an outer automorphism of a free group induced by a braid action to be order-preserving using the reduced Burau representation. This is a work in progress joint with Jonathan Johnson.
Carl-Fredrik Nyberg-Brodda (KIAS)
Title: The growth of some free algebraic objects
Abstract
I will discuss some recent work on counting the growth (in the sense of Milnor) of certain free objects. These free objects – free inverse semigroups, and free regular ⋆-semigroups, respectively – lie somewhere between semigroups and groups, and have a wealth of combinatorial tools and tricks available to them, including rewriting systems. This is in part joint work with M. Kambites, N. Szakács, R. Webb (all in Manchester).
Alan Reid (Rice University / KIAS)
Title: Freiheitssatz, meridional subgroups of knot groups and planar surfaces
Abstract
The classical Freiheitssatz for 1-relator groups says that if G = ⟨x₁, … , xₙ | r = 1⟩ is a 1-relator group with cyclically reduced relator r, and if x₁ appears in r, then the subgroup of G generated by x₂, … , xₙ is a free group (freely generated by x₂, … , xₙ). In this talk we discuss to what extent a "Freiheitssatz type" result holds for subgroups of knot groups generated by meridians. The analysis of subgroups of knot groups generated by finitely meridians then leads to an analysis of (and some interesting questions about) essential planar surfaces with meridional boundary immersed in the knot exterior.
Yandi Wu (Rice University)
Title: Length Spectrum Rigidity Phenomena for Surface Amalgams
Abstract
The length spectrum of a locally CAT(−1) metric space is a multiset of lengths of closed geodesics while the marked length spectrum also specifies the closed geodesic each length is associated with. While celebrated results by Croke and Otal prove that the marked length spectrum determines a metric on a (closed, negatively curved Riemannian) surface, constructions by Vigneras and Sunada show in contrast that the unmarked length spectrum is not enough to determine the metric on a surface. In this talk, I show that similar phenomena hold for negatively curved surface amalgams, natural generalizations of surfaces.