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

## Tuesdays 3:50-4:50. The seminar in Fall 2020 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 Feb. 2nd Feb. 9th Feb. 16th Feb. 23rd March 2nd March 9th March 16th No Seminar Spring break! March 23th March 30th April 6th April 13th April 20th April 27th

## 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.