Schedule of Topology Seminars

In joint work with Qingtao Chen, we consider a family of Turaev-Viro type invariants for a 3-manifold \(M\) with non-empty boundary, indexed by an integer \(r\geq3\), and propose a volume conjecture for hyperbolic \(M\) that these invariants grow exponentially at large \(r\) with a growth rate the hyperbolic volume of \(M\). The crucial step is the evaluation at the root of unity \(\exp(2\pi\sqrt{-1}/r)\) instead of at the usually considered root \(\exp(\pi\sqrt{-1}/r)\). Evaluating at the same root \(\exp(2\pi\sqrt{-1}/r)\), we then conjecture that the original Turaev-Viro invariants and the Reshetikhin-Turaev invariants of a closed hyperbolic 3-manifold \(M\) grow exponentially with the growth rates respectively the hyperbolic and the coplex volume of \(M\). This uncovers a different asymptotic behavior of the values at other roots of unity than that at \(\exp(\pi\sqrt{-1}/r)\) predicted by Witten's Asymptotic Expansion Conjecture, which may indicate some different interpretation of the Reshetikhin-Turaev invariants than the \(SU(2)\) Chern-Simons theory. Numerical evidence will be provided to support these conjectures.

A useful way to study hyperbolic structures on surfaces is to decompose the surface into pairs of pants, understand all the possible hyperbolic structures on a pair of pants, and then understand how such structures can be glued together. In higher dimensions, this technique no longer useful for studying hyperbolic structure. However in this talk we will see that these techniques can be generalized to study convex projective structures on non-hyperbolic manifolds. This work is joint with Jeff Danciger and Gye-Seon Lee.

I'll first give an introduction to Jones' planar algebras, which are a useful tool for the construction and classification of subfactors and fusion categories. A folklore theorem says that sufficiently nice planar algebras are equivalent to pivotal tensor categories together with a distinguished choice of generating object. I'll then discuss joint work with Henriques and Tener, which generalizes the notion of a planar algebra in the category of vector spaces to a planar algebra internal to a modular tensor category C. We generalize the above theorem, showing that planar algebras internal to C are in one-to-one correspondence with module tensor categories M for C, a functor from C to the Drinfel'd center Z(M), and a distinguished object in M which generates M as a C-module.

Thurston showed that 3-manifolds obtained as mapping tori of pseudo-Anosov homeomorphisms on surfaces admit hyperbolic structures, but there is a more natural Sol structure obtained from the structure of the pseudo-Anosov homeomorphism. The Sol structure can be considered as a real projective structure, and I will discuss deformations to nearby projective structures, including recovering hyperbolic cone structures on the manifold.

I will explain how to use TQFT representations of mapping class groups to produce linear representations of surface groups in which every simple closed curve has finite order, but which have infinite image. As a corollary, I will show how to produce covers of surfaces where the integral homology is not generated by pullbacks of simple closed curves on the base. This talk represents joint work with R. Santharoubane.

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact: an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant \(\Sigma(G)\) containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of \(\Sigma(G)\) and \(-\Sigma(G)\) is connected then for every splitting of G as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible. This talk is based on joint work with Spencer Dowdall and Christopher Leininger.

A classical problem in the interplay between geometry and topology is to determine with what types of geometry a fixed manifold can be endowed. The case of surfaces goes back to the late 19th and early 20th centuries through the work of Riemann, Klein, Poincare, and others. One of the seminal results in this area is that every closed surface admits exactly one of three types of homogeneous Riemannian structures that depends only on the sign of its Euler characteristic. There is an analogous result for 3-manifolds, conjectured by Thurston in the 70's and proven by Perelman in '03. However, the statement is not as simple as in dimension 2, as it requires cutting the manifold into pieces, each of which admits a homogenous Riemannian structure. In this talk we will survey the development from the field as well as describe recent results allowing one to "geometrize" certain 3-manifolds without the need to cut them into pieces.

I will outline an extension from closed to finite volume hyperbolic 3-manifolds of a theorem of Kahn-Markovic concerning the existence of sufficient quasi-Fuchsian surfaces to give an virtually special cube complex. Joint with David Futer

The study of outer automorphism group of a free group \(Out(F_N)\) is closely related to the study of Mapping Class Group of a surface. We will discuss various free group analogs of pseudo-Anosov homeomorphisms of hyperbolic surfaces. We will focus mostly on dynamics of their actions on the space of currents and deduce several structural results about subgroups of \(Out(F_N)\).

In 1981 Masur proved the existence of a dense Teichmueller geodesic in moduli space. As an \(Out(F_r)\) analogue, we construct dense geodesic rays in a version of the unit tangent bundle for the quotient of reduced Outer Space, as well as other interesting subcomplexes. The talk will start at an introductory level with a review of Outer Space, Teichmueller space, and the groups \(Out(F_r)\). This is joint work with Yael Algom-Kfir.

The involution (resp. commutator) length of a group G is the smallest integer n, if any, such that any element of G is a product of at most n involutions (resp. commutators). Basmajian and Maskit investigated these lengths in the isometry groups of the constant curvature model spaces, and showed in particular that SO(n,1) has involution length 2 or 3 (depending on the parity of n) and commutator length 1. We study the analogous question for complex hyperbolic isometries, and show that SU(2,1) has involution length 4 and commutator length 1. This is joint work with Pierre Will.

The classical works of Cauchy, Alexandrov and Pogorelov show the existence and uniqueness of compact convex polytopes and convex bodies in Euclidean 3-space. The generalization to convex polytopes in hyperbolic 3-space has generated many exciting results. These include works of Andreev, Epstein-Marden, Fillastre, Rivin, Sullivan, Thurston and others. We will show that there is a close relationship between ideal convex polyhedra in hyperbolic 3-space and a discrete uniformization theorem for polyhedral surfaces which are obtained by gluing Euclidean triangles. Some open questions will be discussed. This is a joint work with D. Gu, J. Sun and T. Wu.

As with SL(2,R) acting on hyperbolic space, a central method for studying a mapping class group is to study its action on its Teichmuller space and a central method for studying an outer automorphism group of a free group \(Out(F_r)\) is to study its action on its Culler-Vogtmann Outer Space \(CV_r\). Each of these groups also have elements acting in some sense "hyperbolically" (pseudo-Anosov elements of mapping class groups and fully irreducible outer automorphisms of free groups). However, the analogy breaks down when one wants to study the invariant axis for a fully irreducible. It appears the correct object to study is actually a collection of axes, an "axis bundle." By proving when the axis bundle for a fully irreducible is just a single axis, we have highlighted the setting where a fully irreducible also behaves in this regard like a pseudo-Anosov or hyperbolic element. This then motivates our study of the cyclic subgroup generated by a fully irreducible outer automorphism with one of these "lone axes." We are able to exactly characterize the centralizer, normalizer, and commensurator of these cyclic subgroups. This is joint work with Yael Algom-Kfir and Lee Mosher.

I will describe a family of 2-bridge knots whose \(SL_2(\mathbb{C})\) character varieties contain multiple curve components of characters of irreducible representations. The intersection points between components carry interesting topological information. In particular, they are associated to Seifert surfaces. Along the way, I will introduce the character variety and describe how certain points are associated to splittings via actions on trees.

Mostow showed the geometry of a finite volume hyperbolic manifold is determined by its loops (fundamental group). Cooper and Delp showed that the geometry of a compact strictly convex projective manifold is determined up to duality by the length of the shortest loops (length spectrum). I will discuss the analogous results for strictly and properly convex projective manifolds in the non-compact setting. The key ideas are that the length spectrum can be computed algebraically and that the Zariski closure of the manifold isometry group is either \(\textrm{SO}(n,1)\) or \(\textrm{SL} (n+1,\mathbb{R})\).

A lattice in SL(3; R) is a discrete subgroup of SL(3; R) that has finite covolume. A subgroup of a lattice in SL(3; R) is thin if it has infinite index in the lattice and is Zariski-dense in SL(3; R). Recently, there has been much interest in thin subgroups of lattices in SL(3; R). In this talk, we discuss some results of Long, Reid, and Thistlethwaite proving the existence of infinite families of thin subgroups of lattices in SL(3; R) arising from representations of hyperbolic triangle groups into SL(3; R). This talk is mostly introductory, but knowledge of covering space theory, hyperbolic surfaces, and representation theory will be helpful.

To any closed orientable surface we can associate a vector space generated by admissible labelings of the spine of the handlebody bounded by the surface. This vector space arises from the Reshetikhin-Turaev TQFT construction. A natural action of the mapping class group of the surface on this vector space arises from the Turaev-Viro construction . We look to use skein theoretic techniques to explore the properties of this representation when the labelings are obtained from SU(3) at a fixed level. In particular we will show that any non-central element of the mapping class group is detected by this representation at a sufficiently high level, meaning the direct sum over all possible levels gives a faithful representation of the mapping class group of any surface.

The Nielsen-Thurston classification of surface automorphisms of a closed orientable surface of genus g states that elements of the mapping class group, Mod(\(S_g\)), are either Periodic, Reducible, or Pseudo-Anosov. There is a rich theory surrounding Pseudo-Anosov mapping classes and in this talk I will discuss two general constructions of Pseudo-Anosov maps and I will discuss the restrictions and differences of these two constructions. This talk is very accessible and no background is required.

A n-sided polygon in \(\mathbb{R}^3\) can be described as a point in \(\mathbb{R}^{3n}\) by listing in order the coordinates of it vertices. In this way, the space of n-sided polygons embedded in \(\mathbb{R}^3\) is a manifold in which points correspond to piecewise linear knots and paths correspond to isotopies which preserve the geometric structure of these knots. Restricting to polygons of unit edge length gives a submanifold consisting of equilateral knots. In addition, the space of equilateral polygons is a toric symplectic manifold. In this talk, we will discuss some aspects of the topology of the space of equilateral hexagons as well as its symplectic structure.

We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called generalized Hopf banding. We further characterize generalized crossing changes between fibered links, and the resulting changes in monodromy. We harness these results to answer a question in molecular biology about classifying DNA unlinking pathways. This is joint work with Matt Rathbun, Kai Ishihara and Koya Shimokawa.