# Geometry and Topology Seminar 2021

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

The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 of Wachman Hall.

• Friday January 29, 2021 at 10:40, over Zoom
Isospectral hyperbolic surfaces of infinite genus

Federica Fanoni, University of Paris

Abstract: Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.

• Friday February 12, 2021 at 10:40,
Constructing examples and non-examples of jigsaw pseudomodular groups

Carmen Galaz-Garcia
University of California, Santa Barbara

Consider a discrete subgroup H of PSL(2,R) and its action on IH^2 the upper half-plane model of the hyperbolic plane. The cusp set of H is the set of points in the boundary at infinity of IH^2 fixed by its parabolic elements. For example, the cusp set of PSL(2,Z) is QU{oo}. A natural question is: how strong is the cusp set as an invariant? More precisely, if H has cusp set  QU{oo} , is it commensurable with PSL(2,Z)? A negative answer was provided by Long and Reid in 2001 by constructing finitely many examples of pseudomodular groups. In 2016 Lou, Tan and Vo produced two infinite families of pseudomodular groups via the jigsaw construction. In this talk we will construct a third family of pseudomodular groups obtained with the jigsaw construction and also show that "many" of the simplest jigsaw groups are not pseudomodular.

• Friday February 19, 2021 at 10:40, over Zoom
A polynomial invariant for veering triangulations

Sam Taylor

Temple University

Veering triangulations form a rich class of ideal triangulations of cusped hyperbolic manifolds that were introduced by Agol and have connections to hyperbolic geometry, Teichmuller theory, and the curve complex. In this talk, we introduce a polynomial invariant of a veering triangulation that, when the triangulation comes from a fibration, recovers the Teichmuller polynomial introduced by McMullen. We show that in general, the polynomial determines a (typically non-fibered) face of the Thurston norm ball and that the classes contained in the cone over this face have representatives that are carried by the veering triangulation itself.

This is joint work with Michael Landry and Yair Minsky.

• Friday February 26, 2021 at 10:40, over Zoom
Cannon-Thurston maps: to exist or not to exist

Radhika Gupta, Temple University

Consider a hyperbolic 3-manifold which is the mapping torus of a pseudo-Anosov on a closed surface of genus at least 2. Cannon and Thurston showed that the inclusion map from the surface into the 3-manifold extends to a continuous, surjective map between the visual boundaries of the respective universal covers. This gives a surjective map from a circle to a 2-sphere. In this talk, we will explore what happens when we consider the mapping torus of a surface with boundary, which is not hyperbolic but CAT(0) with isolated flats.

• Friday March 5, 2021 at 10:40, over Zoom
Geodesic currents and the smoothing property

Didac Martinez-Granado, UC Davis

Abstract: Geodesic currents are measures that realize a closure of the space of curves on a closed surface. Bonahon introduced geodesic currents in 1986, showed that geometric intersection number extends to geodesic currents and realized hyperbolic length of a curve as intersection number with a geodesic current associated to the hyperbolic structure. Since then, other functions on curves have been shown to extend to geodesic currents. Some of them extend as intersection numbers, such as negatively curved Riemannian lengths (Otal, 1990) or word length w.r.t. simple generating sets of a surface group (Erlandsson, 2016). Some other functions aren't intersection numbers but extend continuously (Erlandsson-Parlier-Souto, 2016), such as word length w.r.t. non-simple generating sets or extremal length of curves. In this talk we present a criterion for a function on curves to extend continuously to geodesic currents. This is joint work with Dylan Thurston.

• Friday March 12, 2021 at 11:00, over Zoom
RAAGs in MCGs

Ian Runnels, University of Virginia

Abstract: Inspired by Ivanov's proof of the Tits alternative for mapping class groups via ping-pong on the space of projective measured laminations, Koberda showed that right-angled Artin subgroups of mapping class groups abound. We will outline an alternate proof of this fact using the hierarchy of curve graphs, which lends itself to effective computations and stronger geometric conclusions. Time permitting, we will also discuss some applications to the study of convex cocompact subgroups of mapping class groups.

• Friday March 19, 2021 at 10:40, over Zoom
Cubical geometry in big mapping class groups

Anschel Schaffer-Cohen, University of Pennsylvania

Abstract: Mapping class groups of infinite-type surfaces, also known as big mapping class groups, can be studied geometrically from the perspective of coarsely bounded generating sets. Within this framework, we describe a large family of surfaces--the avenue surfaces without significant genus--and show that the mapping class group of any such surface is quasi-isometric to an infinite-dimensional cube graph. As a consequence, we see that these mapping class groups are all quasi-isometric to each other, and that they are all a-T-menable. Both of these properties are notable in that they are known to fail for mapping class groups of finite-type surfaces.

• Friday April 9, 2021 at 13:00, over Zoom
Covers, curves, and length spectra

Marissa Loving, Georgia Tech

Abstract: In this talk, I will share some of my ongoing work with Tarik Aougab, Max Lahn, and Nick Miller in which we explore the simple length spectrum rigidity of hyperbolic metrics arising from Sunada’s construction. Along the way we give a characterization of equivalent covers (not necessarily regular) in terms of simple elevations of curves, generalizing previous work with Aougab, Lahn, and Xiao.

• Friday April 16, 2021 at 10:40, over Zoom
A polynomial invariant for square triangulations

Aaron Abrams, Washington & Lee University

Abstract: A celebrated theorem of Monsky from 1970 implies that it is impossible to dissect a square into an odd number of triangles of equal area.

Which begs the question:  which (tuples of) areas can be the areas of the triangles in a dissection of a square?

It turns out that for each combinatorial type of triangulation, there is exactly one polynomial relation that is satisfied by the areas of the triangles in every geometric realization of the given triangulation.  I will talk about recent discoveries and current mysteries surrounding this polynomial invariant.

• Wednesday September 8, 2021 at 14:30, Wachman 617
Systoles and cosmetic surgeries

Dave Futer, Temple University

Abstract: The cosmetic surgery conjecture, posed by Cameron Gordon in 1990, is a uniqueness statement that (essentially) says a knot in an arbitrary 3-manifold is determined by its complement $N$. In the past three decades, this conjecture has been extensively studied, especially in the setting where the knot complement $N$ embeds into the 3-sphere. Many different invariants of knots and 3-manifolds have been applied to this problem.

After surveying some of this recent work, I will describe a recent result that uses hyperbolic methods, particularly short geodesics, to reduce the cosmetic surgery conjecture for any particular $N$ to a finite computer search. This is joint work with Jessica Purcell and Saul Schleimer.

• Wednesday September 15, 2021 at 14:30, Wachman 617
Left orderability for surgeries on [1,1,2,2,2j] two-bridge knots

Khánh Lê, Temple University

Abstract: A group is called left-orderable if it admits a total ordering that is invariant under left multiplication. In 3-manifold topology, left orderability is an important concept due to its role in the L-space conjecture. There has been a substantial effort in developing tools to order the fundamental group of rational homology 3-spheres. In a recent work, Xinghua Gao encoded information about hyperbolic $\widetilde{PSL}_2{\mathbb R}$ representations of a one-cusped 3-manifold $M$ in the holonomy extension locus and used it to order intervals of Dehn fillings assuming a strong technical condition of the character variety of $M$. In this talk, we will show how to weaken this condition to a local condition at the non-abelian reducible representation. As an application, we construct left orders on an interval of Dehn fillings on the $[1,1,2,2,2j]$ two-bridge knots.

• Wednesday September 22, 2021 at 14:30, Wachman 617
Right-angled links in thickened surfaces

Rose Kaplan-Kelly, Temple University

Abstract: Traditionally, alternating links, links with a projection diagram that can be given an orientation such that the link's crossings alternate between over- and under-crossings, are studied with alternating diagrams on $S^2$ in $S^3$. In this talk, we will consider links which are alternating on higher genus surfaces $S_g$ in $S_g x I$. We will define what it means for such a link to be right-angled generalized completely realizable (RGCR) and show that this property is equivalent to the link having two totally geodesic checkerboard surfaces, and equivalent to a set of restrictions on the link's alternating projection diagram. We will then use these diagram restrictions to classify RGCR links according to the polygons in their checkerboard surfaces and provide a bound on the number of RGCR links for a given surface of genus g. Along the way, we will answer a question posed by Champanerkar, Kofman, and Purcell about links with alternating projections on the torus.

• Wednesday September 29, 2021 at 14:30, Wachman 617
Decompositions of groups and their quotients using graphs and cell complexes

Thomas Ng, Technion

Abstract:Bass-Serre theory plays an important role in studying manifolds via decompositions along essential submanifolds by graphically encoding their fundamental groups as a combination of more easily understood subgroups and morphisms between them. Subgroups inherently are related to covers of such graphs of groups, but using these tools to geometrically study quotients is more mysterious. I will discuss how certain classes of quotients admit the generalized structure of developable complex of groups. I will go on to mention joint work with Radhika Gupta and Kasia Jankiewicz demonstrating how to use this structure to prove locally uniform exponential growth for certain Artin groups and the Higman group.

• Friday October 1, 2021 at 15:00, Bryn Mawr College, Park Science Center 159
Graphically discrete groups and rigidity

Emily Stark, Wesleyan University

PATCH Seminar (Joint with Bryn Mawr, Haverford, and Penn)

Abstract: Rigidity theorems prove that a group's geometry determines its algebra, typically up to virtual isomorphism. Motivated by interest in rigidity, we study the family of graphically discrete groups. In this talk, we will present rigidity consequences for groups in this family. We will present classic examples as well as new results that imply this property is not a quasi- isometry invariant. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

In the morning background talk (10-11am), I will discuss coarse geometry and present examples of how groups with similar large-scale geometry often share common algebraic features.

• Friday October 1, 2021 at 16:30, Bryn Mawr College, Park Science Center 159
Liouville cobordisms

Emmy Murphy, Princeton University and IAS

PATCH Seminar (Joint with Bryn Mawr, Haverford, and Penn)

Abstract: In this talk we'll discuss some interesting Liouville cobordisms arising in the particular case when the negative boundary is an overtwisted contact manifold. This will center on two independent constructions: concordances in the high-dimensional setting, and cobordisms with high-index (and therefore non-Weinstein) topological type.

In the morning background talk (11:30am-12:30pm), we'll discuss the basics of Liouville manifolds and Weinstein handles. This is a method by which new symplectic manifolds can be constructed from old, using isotropic/Legendrian submanifolds of contact manifolds.

• Wednesday October 6, 2021 at 14:30, Wachman 617
Polynomial invariants of free-by-cyclic groups

Radhika Gupta, Temple University

Abstract: We will first talk about some polynomial invariants for fibered hyperbolic 3-manifolds, namely the Teichmüller polynomial and Alexander polynomial. We will then develop analogous theory for free-by-cyclic groups and explore the relation between the corresponding polynomials. This is based on joint work with Sam Taylor and Spencer Dowdall.

• Thursday October 21, 2021 at 17:15, Zoom talk
Topology of the space of contact structures on the 3-sphere

Yasha Eliashberg, Stanford University

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: I will prove that the space of positive tight contact structures on the 3-sphere is homotopy equivalent to the real projective plane. The talk is based on a joint work with N. Mishachev.

• Thursday October 21, 2021 at 18:15, Zoom talk
Dilatation for random point-pushing pseudo-Anosovs

Tarik Aougab, Haverford College

PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)

Abstract: Choose a homotopically non-trivial curve “at random” on a compact orientable surface- what properties is it likely to have? We address this when, by “choose at random”, one means running a random walk on the Cayley graph for the fundamental group with respect to the standard generating set. In particular, we focus on self-intersection number and the location of the metric in Teichmuller space minimizing the geodesic length of the curve. As an application, we show how to improve bounds (due to Dowdall) on dilatation of a point-pushing pseudo-Anosov homeomorphism in terms of the self-intersection number of the defining curve, for a “random” point-pushing map. This represents joint work with Jonah Gaster.

• Wednesday October 27, 2021 at 14:30, Wachman 617
Geometric triangulations of a family of hyperbolic 3-braids

Barbara Nimershiem, Franklin & Marshall College

Abstract: After constructing topological triangulations of complements of closures of the braids $C^2\sigma_1^p\sigma_2^{-1}$, where $p\geq 1$, I will argue that the constructed triangulations meet a stronger condition: They are actually geometric. The proof follows a procedure outlined by Futer and Gu\'eritaud and uses a theorem of Casson and Rivin. If time allows, I will also discuss possible generalizations of the construction.

• Wednesday November 10, 2021 at 14:30, Wachman 617
Boundaries of fibered faces and limits of stretch factors

Sam Taylor, Temple University

Abstract: Pseudo-Anosov stretch factors are organized by the fibered face theory of Thurston, Fried, and McMullen. In this talk, I’ll explain the precise sense in which this is true, draw lots of pictures that illustrate it, and use this perspective to answer a question of Chris Leininger. Informally, he asks what can limit points of stretch factors coming from a single 3-manifold look like, and we’ll see how they are actually stretch factors of homeomorphisms of "infinite type" surface that are “wrapped-up” inside in the original manifold.

Anything original is joint work with Landry and Minsky.

• Wednesday November 17, 2021 at 14:30, Wachman 617
Train track maps and CTs on graphs of groups

Rylee Lyman, Rutgers University Newark

Abstract: A homotopy equivalence of a graph is a train track map when it sends vertices to vertices and the restriction of any iterate of the map to an edge yields an immersion. (Relative) train track maps were introduced by Bestvina and Handel in 1992; since then they have become one of the main tools in the study of outer automorphisms of free groups. More recently in 2011, Feighn and Handel introduced a stronger kind of relative train track map called a CT and proved their existence for all outer automorphisms after passing to a power. We extend the theory of relative train track maps to graphs of groups with finitely generated, co-Hopfian edge groups and the theory of CTs to free products (that is, graphs of groups with trivial edge groups).

• Friday December 3, 2021 at 09:30, Wachman 617
Hyperbolic links in the tangent bundle

Andrew Yarmola, Princeton University

PATCH Background talk

Abstract: Staring with a background on the geometry and topology of 2- and 3-manifolds, we will introduce hyperbolic 3-manifolds that arise as link complements in the projectivized tangent bundle $PT(S)$ of a surface $S$. Specifically, we will focus on the case where the link is the canonical lift of a family $C$ of smooth curves. When $C$ is filling and in minimal position on $S$, the resulting 3-manifold $M_C$ turns out to be finite-volume and hyperbolic and therefore any invariants of $M_C$ (such as volume, homology, cusp shape and volume, number of tetrahedra in canonical triangulations, etc) are now mapping class group invariants of $C$. Outside of this connection, these links may be of independent interest as they include all Lorenz links and provide an infinite family Legendrian links for the natural contact structure on $PT(S)$.

• Friday December 3, 2021 at 11:00, Wachman 617
Introduction to knot concordance

Allison Miller, Swarthmore College

PATCH Background Talk

We will talk about the basics of knot concordance and answer some of the following questions, depending on audience preferences: What should it mean for a knot to be "simple" from a 4-dimensional perspective? Is there a sensible and interesting definition of when we should think of two knots as being "4-d equivalent". What do classical knot invariants like the Alexander polynomial have to say? What structure can we find or build from a 4-dimensional perspective on knot theory? Can this help us understand the weird world of 4-manifold topology more broadly?

• Friday December 3, 2021 at 15:00, Wachman 617
Invariants of hyperbolic canonical links in tangent bundles

Andrew Yarmola, Princeton University

PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $C$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $M_C = PT(S) \setminus \hat{C}$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hat C$ is the set of tangent lines to $C$. In particular, any invariant of $M_C$ is a mapping class group invariant of the collection $C$. In this talk, we will go over results that explain the behavior and provide coarse bounds on the volume of $M_C$ in terms of topological and geometric properties of the family $C$ . For example, when $C$ is a filling pair of simple closed curves, we show that the volume is coarsely comparable to Weil-Petersson distance between strata in Teichmuller space. Further, we will explain algorithmic methods and tools for building such links and computing invariants.

• Friday December 3, 2021 at 16:30, Wachman 617
Satellite operators and knot concordance

Allison Miller, Swarthmore College

PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn

Abstract: The classical satellite construction associates to any pattern P in a solid torus and companion knot K in the 3-sphere a satellite knot P(K), the image of P when the solid torus is ‘tied into’ the knot K. This operation descends to a well-defined map on the set of (smooth or topological) concordance classes of knots. Many natural questions about these maps remain open: when are they surjective, injective, or bijective? How do they behave with respect to measures of 4-dimensional complexity? How do they interact with additional group or metric space structure on the concordance set?