# Geometry and Topology Seminar 2020

Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.

The Seminar usually takes place on Fridays at 10:40 AM via Zoom (please contact the seminar organizers for the Zoom link), not in Room 617 on the sixth floor of Wachman Hall.

• Wednesday January 22, 2020 at 14:30, Wachman 617
Detecting Geometric Outer Automorphisms

Edgar A. Bering IV
Temple University

Outer automorphisms of a free group are a fundamental example in geometric group theory and low dimensional topology. One approach to their study is by analogy with the mapping class groups of surfaces. This analogy is made concrete by the natural inclusions Mod(S) -> Out(F) that occur whenever S has free fundamental group. Outer automorphisms in the image of these inclusions are called geometric. In 1992, Bestvina and Handel gave an algorithm for deciding when an irreducible outer automorphism is geometric. I will describe current joint work with Yulan Qing and Derrick Wigglesworth to give an algorithm to decide when a general outer automorphism is geometric.

• Wednesday January 29, 2020 at 14:30, Wachman 617
From Hierarchical to Relative Hyperbolicity

The success of Gromov’s coarsely hyperbolic spaces has inspired a multitude of generalizations. We compare the first of these generalizations, relatively hyperbolic spaces, with the more recently introduced hierarchically hyperbolic spaces. We show that relative hyperbolicity can be detected by examining simple combinatorial data associated to a hierarchically hyperbolic space. As an application, we classify when the separating curve graph of a surface is relatively hyperbolic.

• Wednesday February 5, 2020 at 14:30, Wachman 617
An Analog to the Curve Complex for FC Type Artin groups

Rose Morris-Wright Brandeis University

Artin groups are a generalization of braid groups that provide a rich field of examples and counter-examples for many algebraic, geometric, and topological properties. Any given Artin group contains many subgroups isomorphic to other Artin groups, creating a hierarchical structure similar to that of mapping class groups. I generalize and unify the work of Kim and Koberda on right angled Artin groups and the work of Cumplido, Gonzales-Meneses, Gebhardt, and Wiest on finite type Artin groups, to construct a simplicial complex in analogy to the curve complex. I will define this complex, and discuss some properties that this complex shares with the curve complex of a mapping class group.

• Wednesday February 12, 2020 at 14:30, Wachman 617
Smooth Fibrations of the 3-Sphere by Simple Closed Curves

Leandro Lichtenfelz, University of Pennsylvania We show that the moduli space of all smooth fibrations of a 3-sphere by oriented simple closed curves has the homotopy type of a disjoint union of a pair of 2-spheres, which coincides with the homotopy type of the finite-dimensional subspace of Hopf fibrations. In the course of the proof, we present a pair of entangled fiber bundles in which the diffeomorphism group of the 3-sphere is the total space of the first bundle, whose fiber is the total space of the second bundle, whose base space is the diffeomorphism group of the 2-sphere. This is joint work with D. DeTurck, H. Gluck, M. Merling and J. Yang.

• Friday February 28, 2020 at 14:30, Haverford College, room Sharpless S113
Convergence of normalized Betti numbers in nonpositive curvature

Ian Biringer, Boston College

PATCH Seminar, at Haverford College

Abstract: We’ll show that if $X$ is any symmetric space other than 3-dimensional hyperbolic space and $M$ is any finite volume manifold that is a quotient of $X$, then the normalized Betti numbers of M are “testable", i.e. one can guess their values by sampling the complex at random points. This is joint with Abert-Bergeron-Gelander, and extends some of our older work with Nikolov, Raimbault and Samet. The content of the recent paper involves a random discretization process that converts the "thick part" of $M$ into a simplicial complex, together with an analysis of the "thin parts" of $M$. As a corollary, we can prove that whenever $X$ is a higher rank irreducible symmetric space and $M_i$ is any sequence of finite volume quotients of $X$, the normalized Betti numbers of the $M_i$ converge to the "$L^2$-Betti numbers" of $X$.

There will also be a background talk on this topic at 9:30am.

• Friday February 28, 2020 at 16:00, Haverford College, room Sharpless S113
Right-veering open books and the Upsilon invariant

Diana Hubbard, CUNY

PATCH Seminar, at Haverford College

Abstract: Fibered knots in a three-manifold $Y$ can be thought of as the binding of an open book decomposition for $Y$. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

There will also be a background talk on this topic at 11:00am.

• Wednesday March 11, 2020 at 14:30, Wachman 617
Hyperbolic volume and bounded cohomology

James Farre, Yale University

Abstract: A natural notion of complexity for a closed manifold $M$ is the smallest number of top dimensional simplices it takes to triangulate $M$.  Gromov showed that a variant of this notion called simplicial volume gives a lower bound for the volume of $M$ with respect to any (normalized) Riemannian metric.  The heart of his proof factors through the dual notion of bounded cohomology.  I will define bounded cohomology of discrete groups illustrated by some examples coming from computing the volumes of geodesic simplices in hyperbolic space.  Although bounded cohomology is often an unwieldy object evading computation, we give some conditions for volume classes to be non-vanishing in low dimensions.  We then ask, When do higher dimensional volume classes vanish?’’

• Friday September 11, 2020 at 10:40, over Zoom
Non-uniquely ergodic trees in the boundary of Outer space

There exist non-uniquely ergodic arational laminations on a surface of genus $g \geq 2$. That is, there exists an arational lamination which supports two transverse measures that are not scalar multiples of each other. In analogy, one asks if 'arational' trees in the boundary of Outer space support metrics that are not scalar multiples of each other. In this talk, I will first talk about laminations on surfaces. Then we will see some examples of trees in Outer space and understand what it means for a tree to be non-uniquely ergodic. Time permitting, I will describe the construction of a "non-geometric", arational, non-uniquely ergodic $F_n$-tree. This is joint work with Mladen Bestvina and Jing Tao.