**
GTiNY
Geometric Topology in New York
Columbia University, June 2nd-4th, 2023
**

**Program**

We will start on Friday June 2nd at 2pm, and we will end on Sunday June 4th at Noon.

**Speakers**

We will have 7 plenary talks and 10 sectional talks.

**Plenary speakers**

Michelle Chu (University of Minnesota – Twin Cities)

Nathan Dunfield (University of Illinois at Urbana-Champaign)

Dave Futer (Temple University)

Josh Greene (Boston College)

Damian Osajda (University of Copenhagen)

Olga Plamenevskaya (Stony Brook University)

Nicolas Tholozan (École Normale Supérieure, Paris)

**Sectional speakers**

Assaf Bar-Natan (Brandeis University)

Joe Boninger (Boston College)

Surena Hozoori (University of Rochester)

Sam Hughes (University of Oxford)

Rose Kaplan-Kelly (Temple University/George Mason University)

Giuseppe Martone (Yale University)

Eduardo Reyes (University of California Berkeley)

Jacob Russell (Rice University)

Sumeyra Sakalli (University of Arkansas)

Theodore Weisman (University of Michigan)

**Schedule**

**Friday June 2nd**

2-3pm | Futer (Plenary, Room 312) | ||

3-4:30pm | Parallel Sessions | ||

Special Session 1 (Room 203) | Special Session 2 (Room 207) | ||

3:15-3:45pm | Martone | Sakalli | |

3:55-4:25pm | Weisman | Hozoori | |

4:30-5pm | Coffee & Cookies (Room 508) | ||

5-6pm | Plamenevskaya (Plenary, Room 312) |

**Saturday June 3rd**

8:30-9:30am | Bagels & Coffee (Room 508) | ||

9:30-10:30am | Dunfield (Plenary, Room 312) | ||

10:30am-12:35pm | Parallel Sessions | ||

Special Session 3 (Room 203) | Special Session 4 (Room 207) | ||

10:45-11:15am | Reyes | Boninger | |

11:25-11:55am | Hughes | Bar-Natan | |

12:05-12:35pm | Russell | Kaplan-Kelley | |

12:45pm | Group Picture (Outside the Math building, in front of the Lion) | ||

1-2:30pm | Lunch Break | ||

2:30-3:30pm | Chu (Plenary, Room 312) | ||

3:30-4pm | Coffee & Cookies (Room 508) | ||

4-5pm | Tholozan (Plenary, Room 312) |

**Sunday June 4th**

8:30-9:30am | Bagels & Coffee (Room 508) | ||

9:30-10:30am | Greene (Plenary, Room 312) | ||

11:00am-12:00pm | Osajda (Plenary, Room 312) |

**Titles and Abstracts**

**Speaker:** Assaf Bar-Natan (Brandeis University)

**Title:** Hyperbolicity of Big Mapping Class Groups

**Abstract:** In the work of Mann-Rafi, it was shown that mapping class groups of some infinite-type surfaces are boundedly-generated Polish groups, and thus, following the work of Rosendal, admit a well-defined quasi-isometry type. In this talk, I will present some examples of big mapping class groups that are not Gromov hyperbolic, and some that are. This is joint work in progress with Anschel Shaffer-Cohen and Yvon Verberne.

**Speaker:** Joe Boninger (Boston College)

**Title:** Special Alternating Knots and Ribbon Concordance

**Abstract:** A ribbon concordance between knots in the three-sphere is a relation generalizing the notion of a ribbon knot. Recently, Agol proved ribbon concordance gives a partial ordering on the set of knots, affirming a long-open conjecture of Gordon. We will discuss properties of this ordering, and its interaction with the class of alternating knots. This talk presents ongoing, joint work with Josh Greene.

**Speaker:** Michelle Chu (University of Minnesota – Twin Cities)

**Title:** Hyperbolic 4-manifolds and their embedded submanifolds

**Abstract:** The study of embedded surfaces in hyperbolic 3-manifolds has led to several major advances in the fields of geometry, topology, and geometric group theory. In this talk we address the higher dimensional analogue of embedded 3-manifolds in hyperbolic 4-manifolds. In particular, we address the existence of totally geodesic 3-manifolds in small volume hyperbolic 4-manifolds.

**Speaker:** Nathan Dunfield (University of Illinois at Urbana-Champaign)

**Title:** Counting essential surfaces in 3-manifolds

**Abstract:** Counting embedded curves on a surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting surfaces in a 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many essential surfaces of bounded Euler characteristic up to isotopy in an atoroidal 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory, we can characterize not just the rate of growth but show the exact count is a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein.

**Speaker:** Dave Futer (Temple University)

**Title:** In search of the Margulis constant

**Abstract:** For any hyperbolic 3-manifold M, let M^{< ε} be the set of all points in M whose injectivity radius is less than ε/2. A theorem of Kazhdan and Margulis implies that there is a universal constant μ_{3} such that if ε < μ_{3} then M^{< ε} is topologically a disjoint union of cusps and solid tubes. This constant is useful in many finiteness arguments as well as in effective Dehn surgery. The true value of μ_{3} is unknown. The best lower bound, due to Meyerhoff, says that μ_{3} > 0.104.

In this talk, I will describe partial progress towards the exact value of μ_{3}. In particular, I will explain how a symmetric variant of this constant can be computed and how it can be used, in preliminary computations, to suggest that μ_{3} > 0.5. This is
joint work with David Gabai and Andrew Yarmola.

**Speaker:** Josh Greene (Boston College)

**Title:** From ball fillings to cube tilings

**Abstract:** I will describe an obstruction to a rational homology 3-sphere bounding a smooth rational homology 4-ball. It only applies in special cases, and when it does, it takes the form of a lattice embedding condition. We make a conjecture to the effect that the obstruction is complete, and we prove it in a certain extremal case. The proof utilizes the Hajós-Minkowski theorem that every lattice tiling of Euclidean space by cubes contains a pair of cubes that abut along a facet of each. The talk is based on joint work, part with Slaven Jabuka, part with Brendan Owens.

**Speaker:** Surena Hozoori (University of Rochester)

**Title:** Contact and symplectic geometry of Anosov flows

**Abstract:** Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will discuss a contact/symplectic geometric characterization of Anosov flows in dimension 3 which facilitates the use of new geometric tools in the study of such flows.

**Speaker:** Sam Hughes (University of Oxford)

**Title:** Finite quotients of free-by-cyclic groups

**Abstract:** Profinite rigidity asks how much information about a finitely generated residually finite group can be recovered from its set of finite quotients. In this talk we will describe a recent breakthrough of Yi Liu which proves there are at most finitely many finite volume hyperbolic 3-manifolds with the same set of finite quotients. We will then discuss recent joint work with Monika Kudlinska where we prove an analogous result for generic free-by-cyclic groups.

**Speaker:** Rose Kaplan-Kelly (Temple University/George Mason University)

**Title:** Commensurability and arithmeticity of right-angled links in thickened surfaces

**Abstract:** In this talk, we will consider a generalization of alternating links and their complements in thickened surfaces. 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 consider the commensurability classes of RGCR links and find a family of arithmetic RGCR links.

**Speaker:** Giuseppe Martone (Yale University)

**Title:** d-Pleated surfaces and their coordinates

**Abstract:** Thurston introduced pleated surfaces as a powerful tool to study hyperbolic 3-manifolds. An abstract pleated surface is a representation of the fundamental group of a hyperbolic surface into the Lie group PSL(2,C) of orientation preserving isometries of hyperbolic 3-space together with an equivariant map from the hyperbolic plane into hyperbolic 3-space which satisfies additional properties.

In this talk, we introduce a notion of d-pleated surface for representations into PSL(d,C) which is motivated by the theory of Anosov representations. In addition, we give a holomorphic parametrization of the space of d-pleated surfaces via cocyclic pairs, thus generalizing a result of Bonahon.

This talk is based on joint work with Sara Maloni, Filippo Mazzoli and Tengren Zhang.

**Speaker:** Damian Osajda (University of Copenhagen)

**Title:** Locally elliptic actions and nonpositive curvature

**Abstract:** There are numerous questions concerning actions of torsion groups on nonpositively curved spaces. One example is a well-known conjecture stating that subgroups of CAT(0) groups, that is, of groups acting geometrically on CAT(0) spaces, do not contain infinite torsion (every element has finite order) subgroups. Proving this conjecture might be seen as a first step toward establishing the Tits Alternative for CAT(0) groups. Every action of a torsion group on a CAT(0) space is locally elliptic, that is, every element fixes a point.

Generalizing the above conjecture and a number of other related open questions we state the following Meta-Conjecture: Every locally elliptic action of a finitely generated group on a finite dimensional nonpositively curved complex is elliptic, that is, has a global fixed point. I will explain in details motivations for the Meta-Conjecture, some of its consequences, and relations to well-known open problems. I will present the actual conjectures being specifications of the Meta-Conjecture and explain state of the art, focusing on recent results of my collaborators and myself.

The talk is based on joint works with: Karol Duda, Thomas Haettel, Sergey Norin, and Piotr Przytycki.

**Speaker:** Olga Plamenevskaya (Stony Brook University)

**Title:** Links of surface singularities and their fillings

**Abstract:** Given an isolated singular point in a complex surface, its
link is the intersection of the surface with a small sphere centered
at the singular point. The link is a smooth 3-manifold that reflects
the topology of the singularity. The link carries a natural contact
structure which is Stein fillable: possible smoothings of the
singular point, as well as a deformation of its minimal resolution,
provide a family of Stein fillings. We will examine the relation
between these special fillings of algebraic origin (called Milnor
fillings) and more general Stein fillings of the same contact
manifold, and explain different ways to construct and detect
(non-)Milnor fillings. Partly joint with Baykur--Nemethi and with
Starkston.

**Speaker:** Eduardo Reyes (University of California Berkeley)

**Title:** The space of metric structures on hyperbolic groups

**Abstract:** Hyperbolic groups are generalizations of finitely generated free groups and surface groups, and they act geometrically on Gromov hyperbolic spaces. We can pack all these isometric actions in a single space, which extends the classically studied Teichmüller and Outer spaces, but contains many more interesting actions. I will talk about this space and some of its properties when it is equipped with a natural metric resembling Thurston's Lipschitz metric on Teichmüller space. In particular, I will explain how to find many bi-infinite geodesics in this space, which I showed with Stephen Cantrell.

**Speaker:** Jacob Russell (Rice University)

**Title:** Purely pseudo-Anosov subgroups of 3-manifold groups

**Abstract:** Farb and Mosher's convex cocompact subgroups are some of the geometrically, dynamically, and algebraically richest subgroups of the mapping class group. A major open question about these subgroups asks if they are characterized by each element acting with pseudo-Anosov dynamics on the surface. We show the answer is 'yes' when you restrict to subgroups of fibered 3-manifold groups included into the mapping class group via the Birman exact sequence. Joint work with Chris Leininger

**Speaker:** Sumeyra Sakalli (University of Arkansas)

**Title:** Singular fibers in algebraic fibrations of genus two and their monodromy factorizations

**Abstract:** Kodaira classified all singular fibers that can arise in algebraic elliptic fibrations. Later, Ogg, Iitaka and then Namikawa and Ueno gave a classification for genus two fibrations. In this work, we split these algebraic genus two fibrations into Lefschetz fibrations and determine the monodromies. More specifically, we look at four families of hypersurface singularities in C^{3}. Each hypersurface comes equipped with a fibration by genus 2 algebraic curves which degenerate into a single singular fiber. We determine the resolution of each of the singularities in the family and find a flat deformation of the resolution into simpler pieces, resulting in a fibration of Lefschetz type. We then record the description of the Lefschetz fibration as a positive factorization in Dehn twists. This gives us a dictionary between configurations of curves and monodromy factorizations for some singularities of genus 2 fibrations. This is joint work with J. Van Horn-Morris.

**Speaker:** Nicolas Tholozan (École Normale Supérieure, Paris)

**Title:** Topological obstruction to compact Clifford--Klein forms

**Abstract:** Let G be a semisimple Lie group, H a reductive subgroup, and K a maximal compact subgroup of G containing a maximal compact subgroup L of H. I will present a new and powerful obstruction to the existence of compact quotients of the homogeneous space G/H: if G/H admits a compact quotient, then the normal bundle to K/L inside G/H is "fiberwise homotopically trivial". This weak notion of triviality admits various obstructions ranging from Stiefel--Whitney classes to Adams operations in K-theory, which have many new and interesting applications in our setting.

**Speaker:** Theodore Weisman (University of Michigan)

**Title:** Topological stability for (relatively) hyperbolic boundary actions

**Abstract:** A word-hyperbolic group acts by homeomorphisms on its Gromov boundary, and the dynamics of the action often encode detailed geometric information about the group. Work of Sullivan shows that the action is also stable: any small Lipschitz perturbation of this action is conjugate to the standard action. We will discuss joint work with Kathryn Mann and Jason Manning which shows that any small perturbation of the boundary action of a hyperbolic group is semi-conjugate to the standard action. The key technique is to use topological dynamics to construct a coding of points in the boundary which is well-adapted to the action. Time permitting, we will also describe a relative version of this coding, which can be used to prove a similar result for relatively hyperbolic groups.