Lightning Talks

Schedule

Abstracts

Frankie Chan: Finite quotients of Fuchsian groups.

Abstract: In the setting of all finitely generated Fuchsian groups, we answer the problem of determining these groups via their finite quotients. We provide an effective procedure of a result first shown by Bridson--Conder--Reid. This is joint work with Ryan Spitler.

Tam Cheetham-West: Profinitely rigid fibered hyperbolic 3-manifolds

Abstract: A quick look at some new examples of fibered hyperbolic 3-manifolds whose fundamental groups are completely determined by their sets of finite quotients.

Taylor Daniels: Weighted partition numbers and their asymptotics.

Abstract: Following a brief overview of classical partition-theory results due to Hardy, Ramanujan, and others, a class of "weighted" partition numbers is examined.

Nathan Dunfield: Computing a link diagram from its exterior

Abstract: We give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. In Alan's honor, we use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2,500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincare conjecture. This is joint work with Cameron Rudd and Malik Obeidin.

Hasitha Anuradha Ekanayake: Hyperbolic 3-manifolds with non empty totally geodesic boundary and one cusp

Abstract: My talk will be on our ongoing project to identify the minimal volume manifold in the class of hyperbolic 3-manifolds with non empty totally geodesic boundary and one cusp. Some of the results from the work of Kojima and Miamoto in which they dealt with the volume bounds for compact hyperbolic 3- manifolds with non empty totally geodesic boundary will be mentioned and I will talk about how to improve these bounds so that they can be applied in our setting.

Elizabeth Field: Stable commutator length on big mapping class groups

Abstract: In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will see that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. We will also discuss certain topological properties of a class of infinite-type surfaces and their end spaces. This talk represents joint work with Priyam Patel and Alexander Rasmussen.

Mark Fincher: Hyperbolic orbifold canonical decompositions

Abstract: The canonical decomposition of a finite-volume non-compact hyperbolic 3-manifold is a unique polyhedral decomposition determined by the geometry of the manifold. Its combinatorial data is captured as the Isometry Signature, a complete invariant which SnapPea can compute. We discuss how to do these things for finite-volume non-compact hyperbolic 3-orbifolds.

Arshia Gharagozlou: An Orbifold Census

Abstract: We try to describe the collection of orientable orbifolds that are commensurable with the figure-eight knot complement.

Justin Katz: Spectral rigidity of some arithmetic surfaces.

Abstract: It has been known since the 60's that one cannot hear the shape of a drum, in general. That being said, one should expect that special-enough drums produce special-enough sounds to determine their source. In this talk, I will outline my proof that some very special drums (principal congruence covers of certain arithmetic hyperbolic surfaces) are determined by the way they sound.

Grant Lakeland: Systoles of hyperbolic punctured spheres.

Abstract: For genus zero hyperbolic surfaces with n cusps, we bound the systole length for arithmetic examples for certain values of n. In some cases we show that it is possible to find non-arithmetic examples with systoles longer than is possible among arithmetic examples with the same number of cusps. This is joint work with Clayton Young.

Khanh Le: Totally geodesic surfaces in hyperbolic knot complements

Abstract: The study of surfaces has been essential in studying the geometry and topology of the 3-manifolds that contain them. 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 various new techniques to count totally geodesic surfaces in hyperbolic knot complements. We will also present examples of infinitely many non-commensurable (non-arithmetic) hyperbolic 3-manifolds that contain exactly k totally geodesic surfaces for every positive integer k. We will report on ongoing project on counting totally geodesic surfaces in the complement of knot with small crossing number. This is joint work with Rebekah Palmer.

Marissa Loving: End-periodic homeomorphisms

Abstract: In this talk, we will introduce end-periodic homeomorphisms of infinite-type surfaces and discuss some of the ways they mimic the behavior of psuedo-Anosov homeomorphisms in the finite-type setting. For example, their associated mapping tori admit compactifications with well-prescribed geometry under certain irreducibility conditions. Our 5-minutes permitting, we will also consider their action on the so-called omnipresent arc graph.

Rylee Lyman: Lipschitz metric isometries between Outer Spaces of virtually free groups.

Abstract: Dowdall and Taylor observed that given a finite index subgroup of a free group, taking covers yields an embedding from the Outer Space of the free group to the Outer Space of the finite-index subgroup and that moreover this is an isometry with respect to the (asymmetric) Lipschitz Metric. We extend this observation to virtually free groups and discuss some consequences.

Christian Millichap: Flat fully augmented links are determined by their complements.

Abstract: The Gordon-Luecke Theorem states that knots are determined by their complements, that is, if two knot complements are homeomorphic, then these knots are equivalent. While this fact does not hold when we expand to links, it is natural to ask: are certain infinite classes of links determined by their complements? In this talk, we will introduce a large and interesting class of links, called flat fully augmented links, and briefly discuss how the hyperbolic geometry of such links can be exploited to show that they are determined by their complements.

Casandra Monroe: Branched Bending

Abstract: For a hyperbolic n-manifold, bending along a totally geodesic hypersurface is a well studied method of producing deformations of the original hyperbolic structure. In this talk, we will consider a generalization of bending, where instead we bend along totally geodesic branched complexes.

Joao Nogueira: Freiheitsatz on meridional knot groups and immersed planar surfaces

Abstract: The Freiheitssatz for 1-relator groups states that any subgroup generated by a proper subset of the generators is free. We consider a Freiheitssatz type question on meridional knot groups: whether every subgroup generated by a proper subset of minimal meridional generators is free. We discuss this question for knots in general and answer it for Montesinos knots. This talk is based on joint work with A. Reid and C. Gordon.

Rebekah Palmer: Algebras of the Whitehead link complement

Abstract: Let $\Gamma$ be the fundamental group of a knot or link complement. The discrete faithful representation into $PSL_2(\mathbb{C})$ has an associated quaternion algebra. We can zoom out and look at the canonical component of the character variety $X(\Gamma)$. For almost every point on $X(\Gamma)$, there is an associated quaternion algebra whose definition is polynomial in the coordinates of the point. We'll see how this algebra behaves for the Whitehead link complement and how the algebra can descend to quaternion algebras of the (p,q)-surgeries thereon.

Mark Pengitore: Short Laws for finite quotients of the Mapping Class groups

Abstract: We provide the first nontrivial infinite family of short laws for principal congruence quotients associated to finite solvable covers of the associated surface.

Nicholas Rouse: Infinitely many knots with non-integral trace.

Abstract: In joint work with Alan Reid, we show that there are infinitely many distinct hyperbolic knots in the $3$-sphere of the form $\mathbb{H}^3/\Gamma$ for $\Gamma$ a discrete subgroup of $\mathrm{PSL}_2(\mathbb{C})$ such that $\Gamma$ has a element whose trace is an algebraic non-integer.

Lorenzo Ruffoni: Special cubulation of strict hyperbolization.

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure is based on the construction of suitable arithmetic hyperbolic lattices, and it has been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. In joint work with J. Lafont, we construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite.

Connor Sell: Cusps and Commensurability Classes

Abstract: There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. Sometimes we can obstruct particular cusp cross-sections from occurring in a commensurability class of arithmetic hyperbolic 4-manifolds using the theory of quadratic forms.

Ryan Spitler: Congruence Completions of Arithmetic Groups

Abstract: Arithmetic groups have many finite quotients coming from reducing their matrix entries by modular congruences. I will hint at some of the things one can see about an arithmetic group by looking at its congruence quotients.

Em Thompson: A-polynomials and cluster algebras

Abstract: The A-polynomial is a knot invariant that may be calculated from an ideal triangulation of a hyperbolic knot. Meanwhile, a cluster algebra is a commutative ring for which generators are determined inductively. Since their inception in the early 2000s, cluster algebras have been shown to appear in a wide range of contexts. In this talk we will consider a connection between cluster algebras associated to triangulated surfaces, and the calculation of A-polynomials.

Yi Wang: Integrals models of canonical components

Abstract: One may study characteristic p versions of canonical components of S(2,C) character varieties of hyperbolic 3-manifolds. I will survey some previous work in this direction, and introduce potential new ways to study these models, such as Arakelov theory.

Neza Zager Korenjak: Proper affine actions of free groups

Using "higher strip deformations", we can find proper actions of free groups on $R^{4n-1}$.

Yongquan Zhang: Geodesic planes in hyperbolic 3-manifolds of infinite volume

Abstract: How do isometrically immersed totally geodesic planes behave in a hyperbolic 3-manifold? When the 3-manifold has finite volume, any geodesic plane is either closed or dense, due to Ratner and Shah independently. In this talk, we will briefly describe some recent developments for the infinte volume case, especially some new phenomenon that may arise.