**Mostly Teichmuller Spaces (MTS) Seminar **is intended to feature seasoned and young researchers in the fields related to quasiconformal maps, Riemann surfaces, hyperbolic geometry, Teichmuller spaces, or any other related topic. The meetings will be on Zoom. We will meet on some Thursdays 11am-12pm during the semester.

Zoom Link:

https://us02web.zoom.us/j/81303849091?pwd=MSt6RnhTdGpacEhrZVRmS0swaXFEUT09

Meeting ID: 813 0384 9091

Passcode: 746878

## Schedule

## Spring 2022

**Anja Randecker**, Heidelberg University, February 24, 10:45am (EST)**Title:**Examples of infinite rational IETs**Abstract:**Interval exchange transformations (IETs) are simple enough, yet interesting dynamical systems: An interval is cut into finitely many subintervals and their order is interchanged. If the lengths of the subintervals are all rational, the IET is particularly simple as it is periodic. When we want to increase the difficulty of the dynamical questions a bit, we can allow infinitely many subintervals. In my talk, I will explain where you can find examples of such infinite IETs and describe the dynamical properties of one explicit example (based on joint work with Pat Hooper and Kasra Rafi).

**Ferran Valdez**, Centro de Ciencias Matematicas, March 3, 10:45am (EST)**Title:**Veech groups infinite type flat surfaces**Abstract:**We consider, for a fixed (infinite-type) surface S, the problem of determining all possible countable groups Veech groups (i.e. linear parts of affine transformations) that can arise from translation surface structure on S. We will start by recalling basic definitions and then dedicate most of the talk to explain a geometric construction that solves the aforementioned problem. This is joint work in progress with Artigiani, Randecker, Sadanand & Weitze-Schmithuesen.

**Bram Petri**, Sorbonne Université, March 24, 11am (EST)**Title:**Random hyperbolic surfaces

**Abstract:**I will talk about random hyperbolic surfaces: why they are interesting, what we know about them and what we would like to know.

**Nhat Minh Doan**, University of Luxembourg, March 31, 11am (EST)**Title:**On a tree structure of the set of orthogeodesics on hyperbolic surfaces**Abstrac**t: Orthogeodesics are defined as geodesic arcs perpendicular to the boundary of a hyperbolic surface at their end points. The lengths of orthogeodesics on hyperbolic surfaces satisfy equations relating these lengths to geometric and topological quantities of the surface. The Basmajian and Bridgeman identities are two of these equations. This talk will be about a tree structure on the set of orthogeodesics leading to a combinatorial proof of Basmajian’s identity in the case of surfaces. As another application, we apply a recursive formula on the tree to compute the ortho-length spectrum on some special surfaces.

**Maxime Fortier Bourque**, Université de Montréal, May 12, 11am (EST)**Title:**A divergent horocycle in the horoboundary of the Teichmuller metric**Abstract:**There is a lesser-known cousin of the Thurston compactification of Teichmuller space known as the Gardiner-Masur compactification, which uses extremal length instead of hyperbolic length. This compactification turns out to be isomorphic to the horofunction compactification of the Teichmuller metric, a general compactification due to Gromov. While the Thurston compactification is homeomorphic to a closed ball, we do not know the shape of the Gardiner-Masur compactification. In this talk, I will describe an example of a horocycle in Teichmuller space which does not converge in the horofunction compactification. This is analogous to a result of Lenzhen about Teichmuller geodesics that diverge in the Thurston compactification.

## Fall 2021

**Hugo Parlier**, University of Luxembourg, October 14, 11 am (EST)**Title:**Curves, surfaces and intersection**Abstract:**On closed surfaces of positive genus, through classical work of Dehn, simple closed curves can be described using intersection numbers. Now what if you want to describe curves with self-intersections in a similar way? This talk will be on joint work with Binbin Xu about this question, and where we end up constructing and studying so-called k-equivalent curves. These are distinct curves that intersect all curves with k self-intersections the same number of times.

**Alastair Fletcher**, Northern Illinois University, October 21, 11 am (EST)**Title:**Cantor sets and Julia sets**Abstract:**One does not have to study complex dynamics much before coming across examples of Julia sets which are Cantor sets. It is then a natural question to ask which Cantor sets can be Julia sets? The rigidity of holomorphic maps precludes certain examples, and so we will ask this question in the context of uniformly quasiregular mappings with a focus on dimensions two and three. Based on joint work with D. Stoertz (Gustavus Adoplhus College) and V. Vellis (University of Tennessee – Knoxville).

**Nicholas Miller**, Berkeley, November 4, 11 am (EST)**Title:**Big mapping class groups and loxodromic isometries on hyperbolic graphs

**Abstract:**For finite-type surfaces, the Nielsen-Thurston classification tells us that every element of the mapping class group is either finite order, fixes a collection of simple closed curves, or is pseudo-Anosov. A coarser version of this classification is captured by the action of the mapping class group on the curve complex, namely the elements which act loxodromically are precisely the pseudo-Anosovs. For infinite type surfaces, much less is known about analogues of these two classifications. In fact, we currently do not even know many explicit examples of mapping classes acting loxodromically on hyperbolic graphs which do not come from the finite-type setting. In this talk, I will give a broad overview of the area and then present some new examples of “instrinsically infinite type” mapping classes acting loxodromically on an infinite-type analogue of the curve complex. This is joint work with Carolyn Abbott and Priyam Patel.

**Harry Baik**, KAIST, December 2, 10am (EST)**Title:**Reducible normal generators for mapping class groups are abundant**Abstract:**There is this philosophy “small translation length means normal generation” for mapping class group actions. For instance, Lanier-Margalit showed that the pseudo-Anosov elements with small translation length on the Teichmüller space normally generate the mapping class group. We provide a similar criterion for reducible mapping classes. This is a joint work with Dongryul M. Kim and Chenxi Wu.

**Xinlong Dong**, Graduate Center, CUNY, December 7, 10am (EST)**Title:**On complex extension of the Liouville map**Abstract:**This is a Colloquium style talk on the Liouville map.