# Dynamics Seminar 2022-2023

During the Spring 2023 semester, the Dynamics seminar meets in room **2425 of Sterling Hall** on **Mondays** from **2:30pm - 3:20pm**. To sign up for the mailing list send an email from your wisc.edu address to dynamics+join@g-groups.wisc.edu. For more information, contact Paul Apisa, Marissa Loving, Caglar Uyanik, Chenxi Wu or Andy Zimmer.

## Spring 2023

## Spring Abstracts

### Pierre-Louis Blayac

A divisible convex set is a convex, bounded, and open subset of an affine chart of the real projective space, on which acts cocompactly a discrete group of projective transformations. These objects have a very rich theory, which involves ideas from dynamical systems, geometric group theory, (G,X)-structures and Riemannian geometry with nonpositive curvature. Moreover, they are an important source of examples of discrete subgroups of Lie groups; for instance they have links with Anosov representations. In this talk, we will survey known examples of divisible convex sets, and then describe new examples obtained in collaboration with Gabriele Viaggi, of irreducible, non-symmetric, and non-strictly convex divisible convex sets in arbitrary dimensions (at least 3).

### Karen Butt

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

### Elizabeth Field

In this talk, I will discuss the geometry of mapping tori which arise from end-periodic homeomorphisms of infinite-type surfaces. In particular, I will give bounds on the volume of these 3-manifolds in terms of the dynamics of the end-periodic map. I will show how the upper bound on volume utilizes tools of subsurface projections, the geometry of the curve complex, and will present the construction of our model manifold. I will also discuss ongoing work on the lower bound which relies on the machinery of pleated surfaces, suitably adapted to our setting of infinite-type surfaces. This talk represents joint work with Autumn Kent, Heejoung Kim, Chris Leininger, and Marissa Loving (in various configurations).

### Chi Cheuk Tsang

The dilatation of a pseudo-Anosov map is a measure of the complexity of its dynamics. The minimum dilatation problem asks for the minimum dilatation among all pseudo-Anosov maps defined on a fixed surface, which can be thought of as the smallest amount of mixing one can perform while still doing something topologically interesting. In this talk, we present some recent work on this problem with Eriko Hironaka, which shows a sharp lower bound for dilatations of fully-punctured pseudo-Anosov maps with at least two puncture orbits. We will explain some ideas in the proof, including standardly embedded train tracks and Perron-Frobenius matrices.

### Paul Apisa

The moduli space of Riemann surfaces is an orbifold whose points correspond to the ways to endow a surface with a complex structure (or hyperbolic metric if you prefer). This is a rich object whose fundamental group is the mapping class group. It comes equipped with a natural metric, called the Teichmuller metric, which determines an action of geodesic flow on the cotangent bundle. This flow and multiplication by elements of C* combine to form a GL(2,R) action.

However, the closures of these GL(2, R) orbits are mysterious.

Work of Eskin, Mirzakhani, Mohammadi, and Filip implies that every one is an algebraic variety. But, aside from two well-understood constructions (one of which entails considering loci of covers; the second of which only works, so far, in genus at most 6), there are only 3 known families of orbit closures - the (infinite family of) Bouw-Moller examples, the 6 Eskin-McMullen-Mukamel-Wright (EMMW) examples, and 3 so-called “sporadic” examples. Building off of ideas of Delecroix, Rueth, and Wright, I will describe work showing that the Bouw-Moller and EMMW examples are both examples of orbit closures that can be constructed using the representation theory of finite groups. The main idea will be to connect these examples to Hurwitz spaces of G-regular covers of the sphere (for an appropriate finite group G) and apply a construction of Ellenberg for finding endomorphisms of the Jacobians of the corresponding Riemann surfaces. In the end, I will describe a construction that inputs a finite group G and a set of generators of G satisfying a combinatorial condition and outputs a GL(2, R) orbit closure.

No background on dynamics on moduli space, Hurwitz spaces, or Riemann surfaces will be assumed.

### Filippo Mazzoli

Since their introduction by Thurston, equivariant pleated surfaces have been extremely useful in the investigation of the geometry of hyperbolic 3-manifolds and the space of surface groups representations inside PSL(2,C). In this talk we will present a generalization of the notion of pleated surface and its associated shear-bend coordinates that is particularly well suited for the study of closed surface group representations in PSL(d,C), and we will introduce a corresponding extension of Bonahon’s holomorphic shear-bend parametrization to any d ≥ 2. This is joint work with Sara Maloni, Giuseppe Martone, and Tengren Zhang.

### Rose Morris-Wright

CANCELLED

### Carolyn Abbott

The visual boundary of a hyperbolic space is a quasi-isometry invariant that has proven to be a very useful tool in geometric group theory. In particular, there is a well- defined notion of the visual boundary of a hyperbolic group. When one considers CAT(0) spaces, however, the situation is more complicated, because the visual boundary is not a quasi-isometry invariant. Instead, one can consider a certain subspace of the visual boundary, called the (sublinearly) Morse boundary. In this talk, I will describe a new topology on this boundary and use it to show that the Morse boundary with the restriction of the visual topology is a quasi-isometry invariant in the case of (nice) CAT(0) cube complexes. This result is in contrast to Cashen’s result that the Morse boundary with the visual topology is not a quasi-isometry invariant of CAT(0) spaces in general. This is joint work with Merlin Incerti-Medici.

### Samantha Fairchild

Given a discrete subgroup in SL(2,R), we consider discrete orbits of this subgroup under the linear action on the Euclidean plane. For example the orbit SL(2,Z) acting on the first basis vector (1,0) gives a subset of the integer lattice. Understanding the distribution of these discrete subgroups is a venerable problem going back to Gauss with the study of the Gauss circle problem and popularized through the study of the geometry of numbers in the mid to late 1900's. We will give a Siegel--Veech type integral formula for averages of pairs of discrete lattice orbits. The applications of this formula will focus on Veech surfaces, which are certain surfaces given by polygons in the plane with opposite sides identified. A classical example of a translation surface is the torus presented as a unit square in the plane with opposite sides identified. We'll present the general theory through focusing on the examples of the torus and my other favorite translation surface: the Golden L. This talk is based on work with Claire Burrin with a dynamics application by Jon Chaika.

### Jon Chaika

The rel foliation is a foliation defined on strata of translation surfaces with at least two cone singularities. This talk will give a conditional proof of the ergodicity of the rel foliation in every such strata with respect to the natural measure, which is called the Masur-Veech measure. This result is conditional on a measure rigidity result for the upper triangular subgroup of $SL(2,\mathbb{R)$, on products of strata that is work in progress of Brown, Eskin, Filip and Rodriguez-Hertz. The rel ergodicity result strengthens earlier work of McMullen; Calsamiglia, Deroin, and Francaviglia; Hammenstadt; and Winsor. Our methods obtain the ergodicity of Real rel flows as well. From this, by known and fairly general arguments in ergodic theory we obtain that the Real rel flows are mixing of all orders. If time permits, the talk will outline the proof that Real rel flows have entropy 0 with respect to Masur-Veech measure. This is joint work Barak Weiss.

### Mikolaj Fraczyk

The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. This invariant rose to prominence after it was used by Gaboriau to distinguish the orbit equivalence relations of essentially free actions of two free groups on different number of generators. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. In my talk I will report on a joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher ranks lattices in semisimple real or p-adic groups. As a corollary we obtain that the number of generators of index n subgroup of such a group grows like o(n) which implies new state of the art results on the growth of mod-p homology groups. The proof is based on certain miraculous properties of Poisson-Voronoi tessellations of higher rank symmetric spaces that might be of independent interest.

### Tarik Aougab

We introduce the notion of a *geodesic current with corners*, a generalization of a geodesic current in which there are singularities (the "corners") at which invariance under the geodesic flow can be violated. Recall that the set of closed geodesics is, in the appropriate sense, dense in the space of geodesic currents; the motivation behind currents with corners is to construct a space in which *graphs* on S play the role of closed curves. Another fruitful perspective is that geodesic currents reside "at infinity'' in the space of currents with corners, in the sense that their (non-existent) corners have been pushed out to infinity. As an application, we count (weighted) triangulations in a mapping class group orbit with respect to (weighted) length, and we obtain asymptotics that parallel results of Mirzakhani, Erlandsson-Souto, and Rafi-Souto for curves. This represents joint work with Jayadev Athreya.

### Didac Martinez-Granado

Consider a hyperbolic group acting on a real tree. On the one hand, we say that it has the "bounded bactracking property" (BBP) if its orbit map is a one-sided quasi-isometry. This notion was introduced for free groups by Gaboriau-Jaeger-Levitt-Lustig in 1998. On the other hand, we say the action is "small" if it has (virtually) cyclic edge stabilizers. In joint work with Misha Kapovich we show that for a large class of hyperbolic groups, small actions have the BBP. Then we relate this to the problem of extending the translation length of the action to geodesic currents, the space of measures on the square of the Gromov boundary. Our work complements recent results of Cantrell--Oregon-Reyes.

## Fall 2022

## Fall Abstracts

### Jing Tao

By Nielsen-Thurston classification, every homeomorphism of a surface is isotopic to one of three types: finite order, reducible, or pseudo-Anosov. While there are these three types, it is natural to wonder which type is more prevalent. In any reasonable way to sample matrices in SL(2,Z), irreducible matrices should be generic. One expects something similar for pseudo-Anosov maps. In joint work with Erlandsson and Souto, we define a notion of genericity and show that pseudo-Anosov maps are indeed generic. More precisely, we consider several "norms" on the mapping class group of the surface, and show that the proportion of pseudo-Anosov maps in a ball of radius r tends to 1 as r tends to infinity. The norms can be thought of as the natural analogues of matrix norms on SL(2,Z).

### Rebekah Palmer

Studying totally geodesic surfaces has been essential in understanding the geometry and topology of hyperbolic 3-manifolds. Recently, Bader--Fisher--Miller--Stover showed that containing infinitely many such surfaces compels a manifold to be arithmetic. We are hence interested in counting totally geodesic surfaces in hyperbolic 3-manifolds in the finite (possibly zero) case. In joint work with Khánh Lê, we expand an obstruction, due to Calegari, to the existence of these surfaces. On the flipside, we prove the uniqueness of known totally geodesic surfaces by considering their behavior in the universal cover. This talk will explore this progress for both the uniqueness and the absence.

### Beibei Liu

The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents.

### Grace Work

The shrinking target problem characterizes when there is a full measure set of points that hit a decreasing family of target sets under a given flow. This question is closely related to the Borel Cantilli lemma and also gives rise to logarithm laws. We will examine the discrete shrinking target problem in a general and then more specifically in the setting of Teichmuller flow on the moduli space of unit-area quadratic differentials.

### Jean Pierre Mutanguha

The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan Canonical Form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!

### Anthony Sanchez

Platonic solids have been studied for thousands of years. By unfolding a platonic solid we can associate to it a translation surface. Interesting information about the underlying platonic solid can be discovered in the cover where more (dynamical and geometric) structure is present. The translation covers we consider have a large group of symmetries that leave the global composition of the surface unchanged. However, the local structure of paths on the surface is often sensitive to these symmetries. The Kontsevich-Zorich mondromy group keeps track of this sensitivity.

In joint work with R. Gutiérrez-Romo and D. Lee, we study the monodromy groups of translation covers of some platonic solids and show that the Zariski closure is a power of SL(2,R). We prove our results by finding generators for the monodromy groups, using a theorem of Matheus–Yoccoz–Zmiaikou that provides constraints on the Zariski closure of the groups (to obtain an "upper bound"), and analyzing the dimension of the Lie algebra of the Zariski closure of the group (to obtain a "lower bound").

### Feng Zhu

Putting hyperbolic metrics on a finite-type surface S gives us linear representations of the fundamental group of S into PSL(2,R) with many nice geometric and dynamical properties: for instance they are discrete and faithful, and in fact stably quasi-isometrically embedded.

In this talk, we will introduce (relatively) Anosov representations, which generalise this picture to higher-rank Lie groups such as PSL(d,R) for d>2, giving us a class of (relatively) hyperbolic subgroups there with similarly good geometric and dynamical properties.

This is mostly joint work with Andrew Zimmer.

### Ethan Farber

A pseudo-Anosov (pA) is a homeomorphism of a compact connected surface S that, away from a finite set of points, acts locally as a linear map with one expanding and one contracting eigendirection. Ubiquitous yet mysterious, pAs have fascinated low-dimensional topologists and dynamicists for the past forty years. We show that any pA on the sphere whose associated quadratic differential has at most one zero, admits an invariant train track whose expanding subgraph is an interval. Concretely, such a pA has the dynamics of an interval map. As an application, we recover a uniform lower bound on the entropy of these pAs originally due to Boissy-Lanneau. Time permitting, we will also discuss potential applications to a question of Fried. This is joint work with Karl Winsor.

### Lukas Geyer

Recently there has been an increased interest in complex dynamics of orientation-reversing maps, in particular in the context of gravitational lensing and as an analogue of reflection groups in Sullivan's dictionary between Kleinian groups and dynamics of (anti-)rational maps. Much of the theory parallels the orientation-preserving case, but there are some intriguing differences. In order to deal with the post-critically finite case, we study anti-Thurston maps (orientation-reversing versions of Thurston maps), and prove an orientation-reversing analogue of Thurston's topological classification of post-critically finite rational maps, as well as the canonical decomposition of obstructed maps, following Pilgrim and Selinger. Using these tools, we obtain a combinatorial classification of critically fixed anti-Thurston maps, extending a recently obtained classification of critically fixed anti-rational maps. If time allows, I will explain applications of this classification to gravitational lensing. Most of this is based on joint work with Mikhail Hlushchanka.

### Harry Baik

I will give a brief introduction to laminar groups which are groups of orientation-preserving homeomorphisms of the circle admitting invariant laminations. The term was coined by Calegari and the study of laminar groups was motivated by work of Thurston and Calegari-Dunfield. We present old and new results on laminar groups which tell us when a given laminar group is either fuchsian or Kleinian. This is based on joint work with KyeongRo Kim and Hongtaek Jung.

### MurphyKate Montee

Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find that properties have a threshold density above which the property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have been well studied are cubulation (and relaxations of this property) and Property (T). In this setting these are mutually exclusive properties, but threshold densities are not known for either property. In this talk I'll present the current state of the art regarding these properties in random groups, and discuss some ways to further these results.

### Marissa Loving

A fundamental question in geometry is the extent to which a manifold M is determined by its length spectrum, i.e. the collection of lengths of closed geodesics on M. This has been studied extensively for flat, hyperbolic, and negatively curved metrics. In this talk, we will focus on surfaces equipped with a choice of hyperbolic metric. We will explore the space between (1) work of Otal (resp. Fricke) which asserts that the marked length spectrum (resp. marked *simple* length spectrum) determines a hyperbolic surface, and (2) celebrated constructions of Vignéras and Sunada, which show that this rigidity fails when we forget the marking. In particular, we will consider the extent to which the unmarked simple length spectrum distinguishes between hyperbolic surfaces arising from Sunada’s construction. This represents joint work with Tarik Aougab, Max Lahn, and Nick Miller.

### Tina Torkaman

In this talk, I will talk about the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

## Archive of past Dynamics seminars

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021