# Geometry and Topology Seminar

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.

• Friday November 20, 2020 at 10:40, over Zoom
Random hyperbolic geometry

Bram Petri, Sorbonne Université

Abstract: In graph theory, random constructions are a classical subject of study. Besides giving one an idea of what a "typical" graph on a large number of vertices looks like, these constructions have also been used to solve certain extremal problems. That is, sometimes it is easier to prove that a random graph has extremal properties than to explicitly construct graphs with these properties. This last idea is often called the probabilistic method. In this talk, I will speak about how these methods can be applied in (mostly low-dimensional) hyperbolic geometry.

• Friday November 6, 2020 at 10:40, over Zoom
Infinitely many virtual geometric triangulations

David Futer, Temple University

Abstract: Every cusped hyperbolic 3-manifold should admit a decomposition into a union of positively oriented ideal tetrahedra. Somewhat shockingly, the question of whether such a geometric triangulation exists is still open. Luo, Schleimer, and Tillmann proved that geometric ideal triangulations of this sort exist in some cover of every cusped 3-manifold. We extend their result by showing that (almost) every cusped hyperbolic 3-manifold has a single cover admitting an infinite sequence of geometric ideal triangulations. The proof involves double coset separability of peripheral subgroups. This is joint work with Neil Hoffman.

• Friday October 23, 2020 at 10:00, over Zoom
Covers from simple closed curves

Tarik Aougab, Haverford College

Abstract: In joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao, we showed recently that a regular cover of surfaces is determined by which closed curves on the base surface admit simple elevations to the cover. This theorem was motivated by a question related to pairs of isospectral hyperbolic surfaces arising from Sunada’s construction, but the assumption of regularity prevented the result from being applicable in that context. In joint work with Max, Marissa, and Nick Miller, we are now able to drop this assumption. We’ll discuss the motivation, some of the ideas used in the arguments, and where there is still work to do.

• Friday October 16, 2020 at 10:40, over Zoom
Extensions of word hyperbolic groups that satisfy a quantitative law alternative

Thomas Ng, Technion

Abstract: Given two groups we can look for a new group containing the first group as a normal subgroup so that the quotient is the second group. Such groups are called group extensions, and arise naturally when studying fiber bundles. I will describe work in progress with Robert Kropholler and Rylee Lyman on how hyperbolicity of the normal subgroup can be leveraged to show that the extension inherits a strengthened quantitative Tits alternative from the quotient group. Along the way we will show that any nonelementary subgroup of the fundamental group of a fibered 3-manifold whose fiber has negative Euler characteristic contains a 6-short free subgroup.

• Friday October 9, 2020 at 10:40, over Zoom
Geometric and group theoretic perspectives on curve intersection

Edgar Bering, Technion

Abstract: Given two simple closed curves a and b on a surface, their geometric intersection number is the minimum number of intersection points over all pairs of curves homotopic to (a, b). This number can be characterized combinatorially, by describing the complementary regions of the two curves, or with the help of hyperbolic geometry. In this talk I will introduce a different point of view, characterizing both simple curves and their intersections group-theoretically. I will then discuss the generalization of the group theoretic point of view to settings far removed from surfaces.

• Friday October 2, 2020 at 10:40, over Zoom
Totally geodesic surfaces in twist knot complements II

Rebekah Palmer, Temple University

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In this talk, we will discuss an infinite family of twist knot complements containing exactly 1 totally geodesic surface, and how to use covering space theory to extend this result to present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly $k$ totally geodesic surfaces for every positive integer $k$. This is joint work with Khanh Le.

• Friday September 25, 2020 at 10:40, over Zoom
Totally geodesic surfaces in twist knot complements I

Khánh Lê, Temple University

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. In particular, there has been considerable work in understanding the existence of totally geodesic surfaces in hyperbolic 3-manifolds. Most recently, Bader, Fisher, Miller, and Stover showed that having infinitely many maximal totally geodesic surfaces implies that the 3-manifold is arithmetic. In a joint work with Rebekah Palmer, we present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly $k$ totally geodesic surfaces for every positive integer $k$. In this talk, I will start with some background in hyperbolic geometry and discuss some ideas behind the proof of the main results. I will also talk about applications of these ideas in showing that a family of twist knot complements is not right-angled and examples of maximal Fuchsian subgroups of infinite covolume in hyperbolic 3-manifolds.

• Friday September 18, 2020 at 10:40, over Zoom
Graphs of curves and arcs quasi-isometric to big mapping class groups

Anschel Schaffer-Cohen, University of Pennsylvania

Abstract: When is a big mapping class group quasi-isometric to a graph whose vertices are curves or arcs on the underlying surface? I will describe a classification of those surfaces whose mapping class groups admit a quasi-isometry to a graph of curves, and provide a specific example of a big mapping class group quasi-isometric a graph of arcs. As a bonus, I will show that the mapping class group of a plane minus a Cantor set is Gromov-hyperbolic. This talk should be accessible to anyone comfortable with geometric group theory and surface topology.

• 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.

• 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 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.

• 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.

• 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.

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