# Rutgers Geometry/Topology seminar: Fall 2020 - Spring 2021

## Tuesdays 3:50-4:50. The seminar in Fall 2020 and Spring 2021 will be held virtually.

Past seminars: 2019-2020, 2018-2019, 2017-2018

## Spring 2021

 Date Speaker Title (click for abstract) Jan. 19th No seminar Jan. 26th Mehdi Yazdi (Oxford) The fully marked surface theorem Feb. 2nd Paul Kirk (Indiana) Calculation of the holonomy perturbed flat SU(2) moduli space of the earring tangle Feb. 9th Khanh Le (Temple) Totally geodesic surfaces in twist knot complements Feb. 16th Xiaolong Hans Han (UIUC) Harmonic forms and norms on cohomology of non-compact hyperbolic 3-manifolds Feb. 23rd Xiaoping Zhu (Rutgers) Convergence of discrete uniformization factor to smooth uniformization function on closed surface of genus greater than 1 March 2nd Hannah Turner (UT Austin) Branched cyclic covers and L-spaces March 9th Inkang Kim (KIAS) Signature, Toledo invariant and surface group representations in SP(2n,R) March 16th No Seminar Spring break! March 23th Sergio Fenley (Florida State) Partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds March 30th Huai-Dong Cao (Lehigh) Rigidity of gradient steady Ricci solitons April 6th Oğuz Şavk (Boğaziçi University) Classical and new plumbings bounding contractible manifolds and homology balls April 13th Samantha Allen (Dartmouth) Using surgery to study unknotting with a single twist April 20th April 27th Daniel Groves (UIC)

## Fall 2020

R
 Date Speaker Title (click for abstract) Sep. 1st No seminar Sep. 8th No seminar Run Monday schedule Sep. 15th Yanwen Luo (Rutgers) The space of geodesic triangulations on surfaces Sep. 22nd Dragomir Saric (CUNY) The Heights theorem for integrable quadratic differentials on infinite Riemann surfaces Sep. 29th Tian Yang (TAMU) A relative version of the Turaev-Viro invariants and hyperbolic polyhedral metrics Oct. 6th No seminar Oct. 13th Lvzhou Chen (UT Austin) Stable commutator lengths of integral chains in right-angled Artin groups Oct. 20th Shaosai Huang (University of Wisconsin Madison) Topological rigidity of the first Betti number and Ricci flow smoothing Oct. 27th Ryan Spitler (McMaster) Profinite Completions of 3-manifold groups and Hyperbolic Triangle Groups Nov. 3rd Jiayin Pan (UCSB) On the escape rate of an open manifold with nonnegative Ricci curvature Nov. 10th No Seminar Nov. 17th Michael Landry (WUSTL) Thurston's norm, veering triangulations, and a new polynomial invariant Nov. 24th No Seminar Happy Thanksgiving! Dec. 1st Tye Lidman (North Carolina State University) SU(2) representations for toroidal homology spheres Dec. 8th

## Abstracts

### The space of geodesic triangulations on surfaces

In 1959, Smale showed that the group of diffeomorphisms of the closed 2-disk which are identity on the boundary is a contractible space. In 1984, Bloch, Connelly, and Henderson proved the discrete version of this theorem: the space of simplexwise linear homeomorphisms of a convex polygon is contractible.

In this talk, we will show that the idea in Tutte's Embedding Theorem can be applied to give a simplified proof of the Bloch-Connelly-Henderson theorem. We will also discuss the homotopy types of the corresponding spaces when the boundary polygon is not convex. Finally, we will discuss the conjecture about the topology of the spaces of geodesic triangulations on closed surfaces.

### The Heights theorem for integrable quadratic differentials on infinite Riemann surfaces

An integrable holomorphic quadratic differential on a Riemann surface induces a measured foliation of the surface by horizontal trajectories. A quadratic differential associates to each homotopy class of a closed curve its height, i.e.-the infimum of the transverse measure over the homotopy class. Marden and Strebel proved that the space of quadratic differentials is in a one to one correspondence to the heights maps when the Riemann surface is of parabolic type.

We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric.

When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded. We also give a sufficient condition for the non-integrable holomorphic quadratic differential to give rise to bounded measured laminations.

### A relative version of the Turaev-Viro invariants and hyperbolic polyhedral metrics

We define a relative version of the Turaev-Viro invariants for an ideally triangulated compact 3-manifold with non-empty boundary and a coloring on the edges, and propose the Volume Conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric with singular locus the edges and cone angles determined by the coloring. As the main result of this talk, we prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the Volume Conjecture for the Turaev-Viro invariants of hyperbolic 3-manifold with totally geodesic boundary.

### Stable commutator lengths of integral chains in right-angled Artin groups

The stable commutator length (scl) is an invariant for group elements, and it carries topological and dynamical information. Topologically, an integral 1-chain in a group G is a formal sum of loops in the K(G,1) space, and its scl is the least complexity of surfaces bounding the union of these loops. Aiming for effective lower bounds of the index of special subgroups (say in hyperbolic 3-manifold groups), we give lower bounds for scl of integral chains in right-angled Artin groups (RAAGs) and right-angled Coxester groups (RACGs). We show that the infimal positive scl of integral chains in any RAAG/RACG is positive, and its size explicitly depends on the defining graph of the RAAG/RACG up to a multiplicative constant 12. In particular, the size is non-uniform among RAAGs/RACGs, but it is uniform among *hyperbolic* RACGs. If time permits, I will also explain a connection between scl in RAAGs and the so-called fractional stability number of graphs. This is joint work with Nicolaus Heuer.

### Topological rigidity of the first Betti number and Ricci flow smoothing

The infranil fiber bundle is a typical structure appeared in the collapsing geometry with bounded sectional curvature. In this talk, I will discuss a topological condition on the first Betti numbers that guarantees a torus fiber bundle structure (a special type of infranil fiber bundle) for collapsing manifolds with only Ricci curvature bounded below. The main technique applied here is smoothing by Ricci flows. This covers my joint with Bing Wang.

### Profinite Completions of 3-manifold groups and Hyperbolic Triangle Groups

The profinite completion of a group encodes the information of all the group's finite quotients. For the fundamental group of a manifold, this is related to the collection of all the manifold's finite covers. A recent object of study has been to investigate to what extent a manifold, for example a 3-manifold, is determined by the profinite completion of its fundamental group. I will discuss the relationship between the profinite completion and the representation theory of a group, and how this relates to recent work establishing that the fundamental groups of certain 3-manifolds and certain hyperbolic triangle groups are determined up to isomorphism by their profinite completions. This represents joint works with Martin Bridson, Ben McReynolds, and Alan Reid.

### On the escape rate of an open manifold with nonnegative Ricci curvature

We study the fundamental groups of open n-manifolds with nonnegative Ricci curvature. They are known to be virtually nilpotent in general. We introduce a quantity, escape rate, that measures how fast the representing geodesic loops escape from bounded balls. We show that if the escape rate is less than some positive constant epsilon(n), then the fundamental group is virtually abelian.

### Thurston's norm, veering triangulations, and a new polynomial invariant

I will describe two related papers concerning the Thurston norm on homology. This norm is a 3-manifold invariant with connections to many areas: geometric group theory, foliation theory, Floer theory, and more. There are some beautiful clues due to Thurston, Fried, Mosher, McMullen, and others that indicate there should be a dictionary between the combinatorics of the norm's polyhedral unit ball and the geometric/topological structures existing in the underlying manifold. However, this picture is incomplete and is mostly limited to the case when the manifold is a surface bundle over the circle. I will explain some results which go beyond the surface bundle case to the more general setting of manifolds admitting veering triangulations, which are combinatorial objects I will define. First I will explain that one of these objects always cuts out the cone over a face of the norm ball in a natural way and computes the norm in this cone. Second, I will explain how one can use these objects to collate all surfaces representing classes in these cones which are compact leaves of taut foliations. Third, I will describe joint work with Yair Minsky and Samuel Taylor in which we use a veering triangulation to define a polynomial invariant, the "veering polynomial," generalizing the Teichmuller polynomial of McMullen and answering a question asked by Calegari and McMullen. This invariant can be obtained by computing the Perron polynomial of a certain directed graph.

### SU(2) representations for toroidal homology spheres

The three-dimensional Poincare conjecture shows that any closed three-manifold other than the three-sphere has non-trivial fundamental group. A natural question is how to measure the non-triviality of such a group, and conjecturally this can be concretely realized by a non-trivial representation to SU(2). We will show that the fundamental groups of three-manifolds with incompressible tori admit non-trivial SU(2) representations. This is joint work with Juanita Pinzon-Caicedo and Raphael Zentner.

### The fully marked surface theorem

In his seminal 1976 paper, Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. We give a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation G such that S is homologous to a union of compact leaves and such that the plane field of G is homotopic to that of F. In particular, F and G have the same Euler class. This is joint work with David Gabai.

### Calculation of the holonomy perturbed flat SU(2) moduli space of the earring tangle

In collaboration with Cazassus, Herald, and Kotelskiy, we identify the flat SU(2) traceless moduli space of a certain tangle introduced by Kronheimer Mrowka to ensure admissibility of bundles in singular instanton homology, and discuss some consequences of the calculation in Floer theory.

### Totally geodesic surfaces in twist knot complements

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 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 a joint work with Rebekah Palmer.

### Harmonic forms and norms on cohomology of non-compact hyperbolic 3-manifolds

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will also describe a partial generalization to the infinite volume case.

### Convergence of discrete uniformization factor to smooth uniformization function on closed surface of genus greater than 1

We investigate the convergence of discrete conformal metrics to the classical uniformization metric. In this talk, we will show that for triangular meshes under certain conditions on a closed surface S of genus greater than 1, there exists an unique discrete uniformization factor such that its difference to the smooth uniformization factor is controlled by the L infinity NORM of the edge length of triangulation. This is a joint work with Tianqi Wu.

### Branched cyclic covers and L-spaces

A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cyclic cover of a knot in the 3-sphere give many examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. This is joint work with Ahmad Issa.

### Signature, Toledo invariant and surface group representations in SP(2n,R)

Recently a great deal of attention has been given to the space of surface group representations utilizing various geometric structures, and invariants. In this talk, we present a new approach to this problem, using Atiyah-Patodi-Singer index theory for the surface with boundary. We will calculate the signature formula in this context, and relate it to the Toledo invariant when the target group is a real symplectic group Sp(2n,R). We also present a version of Milnor-Wood type inequality in this context. This is a joint work with Pierre Pansu and Xueyuan Wan.

### Partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

We analyze partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and show that up to iterates they are either i) discretized Anosov flows, or ii) a double translation example. Case i) means that there is a topological Anosov flow in the manifold M so that the diffeomorphism is a variable time map of this flow. In particular this implies that the diffeomorphism is dynamically coherent. We do a further analysis of the second case and prove geometric properties in a finite cover and for an iterate. This geometric properties imply that the center leaves can be obtained by collapsing flow lines of an associated topological Anosov flow. Together these results imply that if M^3 hyperbolic admits a partially hyperbolic diffeomorphism then it admits an Anosov flow.

A gradient Ricci soliton is a complete Riemannian manifold (M, g), together with a smooth potential function f, such that its Ricci tensor satisfies the equation $Ric + Hess f = \lambda g$, for some constant \lambda. Ricci solitons are important geometric objects because they are natural extensions of Einstein metrics and they model singularity formations in the Ricci flow.

The classification of locally conformally flat (LCF) gradient shrinking Ricci solitons was completed around 2009, and the rigidity of shrinking solitons with harmonic Weyl curvature was proved during 2011-2013 by the combined works of Fernandez-Lopez & Garcia-Rio [2011] and Munteanu & Sesum [2013]. However, for steady solitons, while the LCF ones were classified in 2012, the rigidity problem for steady solitons has been open until very recently. In this talk, I shall discuss the rigidity of gradient steady Ricci solitons and report on my joint work with Jiangtao Yu, a Ph.D student, and also the work of another Ph.D student, Fengjiang Li.

### Classical and new plumbings bounding contractible manifolds and homology balls

A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results together. Our approach is based on Mazur’s famous argument which provides a unification of all results in a fairly simple way.

### Using surgery to study unknotting with a single twist

Ohyama showed that any knot can be unknotted by performing two full twists, each on a set of parallel strands. We consider the question of whether or not a given knot can be unknotted with a single full twist, and if so, what are the possible linking numbers associated to such a twist. It is observed that if a knot can be unknotted with a single twist, then some surgery on the knot bounds a rational homology ball. Using tools such as classical invariants and invariants arising from Heegaard Floer theory, we give obstructions for a knot to be unknotted with a single twist of a given linking number. In this talk, I will discuss some of these obstructions, their implications (especially for alternating knots), many examples, and some unanswered questions. This talk is based on joint work with Charles Livingston.