# Dynamics Seminar 2021-2022

The Dynamics seminar meets in room **B309 of Van Vleck 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 Caglar Uyanik. For VIRTUAL talks, please email Caglar for a zoom link.

## Spring 2022

## Spring Abstracts

### Camille Horbez (Paris-Saclay)

*Measure equivalence rigidity of [math]\displaystyle{ Out(F_n) }[/math]*

Abstract: Measure equivalence between countable groups was introduced by Gromov as a measure-theoretic analogue of quasi-isometry between finitely generated groups. It is related to orbit equivalence of ergodic group actions, and to lattice embeddings on the geometric side. We prove that whenever n is at least 3, the group [math]\displaystyle{ Out(F_n) }[/math] of outer automorphisms of a free group of rank n is superrigid for measure equivalence, i.e. every countable group which is measure equivalent to [math]\displaystyle{ Out(F_n) }[/math] is in fact virtually isomorphic to [math]\displaystyle{ Out(F_n) }[/math]. This is joint work with Vincent Guirardel.

### Andrew Zimmer (UW Madison)

*Entropy rigidity for cusped Hitchin representations*

In this talk I’ll describe an entropy rigidity theorem for certain types of linear representations of geometrically finite Fuchsian groups. The result informally states that the simplest types of representations can be characterizing via their asymptotic behavior. This is joint work with Dick Canary and Tengren Zhang.

### Stephen Cantrell (U Chicago)

*Counting limit theorems for representations of Gromov-hyperbolic groups*

Abstract: The study of random matrix products is concerned with understanding how, when we randomly multiply matrices, natural quantities related to this product (such as the norm or spectral radius) grow and distribute. Of course, we could induce randomness in a variety of different ways: we could multiply matrices following a random walk or according to a Markov chain, for example. During this talk we'll consider a notion of randomness (that in some sense isn't really random!) that takes into account the combinatorial/algebraic structure of the group generated by the matrices that we multiply. We consider a representation of a hyperbolic group G, which we equip with a finite generating set S, into a general linear group and ask: given a group element in G which is chosen uniformly at random from the sphere of radius n in the Cayley graph for (G,S), what should we expect the norm of its corresponding matrix to be? This is based on joint work with Cagri Sert.

### Spencer Dowdall (Vanderbilt)

*Orientability for fully irreducible free group automorphisms*

Abstract: It is well-known that the invariant foliations of a pseudo-Anosov surface homeomorphism are transversely orientable if and only if the the pseudo-Anosov dilatation is equal to the spectral radius of its action on homology. This talk will explore this phenomenon in the context of fully irreducible free group automorphisms. Specifically, we show that equality of geometric and homological stretch factors is a dynamical reflection of the existence of an invariant orientation both on any train track representative and on the expanding lamination of the automorphism. Using this, we obtain an an explicit formula relating the Alexander and McMullen polynomials of the associated free-by-cyclic group, and moreover study the extent to which the property of having an orientable monodromy persists for other free-by-cyclic splittings of the group. Joint work with Radhika Gupta and Samuel J. Taylor.

### Rohini Ramadas (Warwick)

* Special loci in the moduli space of self-maps of projective space *

A self-map of P^n is called post critically finite (PCF) if its critical hypersurface is pre-periodic. I’ll give a survey of many known results and some conjectures having to do with the locus of PCF maps in the moduli space of self-maps of P^1. I’ll then present a result, joint with Patrick Ingram and Joseph H. Silverman, that suggests that for n≥2, PCF maps are comparatively scarce in the space of self-maps of P^n. I’ll also mention joint work with Rob Silversmith, and work-in-progress with Xavier Buff and Sarah Koch, on loci of “almost PCF” maps of P^1.

### Matt Clay (Arkansas)

*Minimal volume entropy of free-by-cyclic groups and 2–dimensional right-angled Artin groups*

Abstract: Let G be a free-by-cyclic group or a 2–dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to G has minimal volume entropy equal to 0. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2–dimensional groups with uniform uniform exponential growth. This is joint work with Corey Bregman.

### Chris Leininger (Rice)

*Purely pseudo-Anosov subgroups of fibered 3-manifold groups*

Abstract: Farb and Mosher, together with work of Hamenstädt, proved that Gromov hyperbolicity for surface group extensions is entirely encoded by algebraic and geometry properties of the monodromy into the mapping class group. They were thus able to give a purely geometric formulation for Gromov’s Coarse Hyperbolization Question for the class of surface group extensions: Given a finitely generated, purely pseudo-Anosov (free) subgroup of the mapping class group, is it convex cocompact? In this talk, I will explain joint work with Jacob Russell in which we answer the question affirmatively for a new class of subgroups, namely, subgroups of fibered 3-manifold groups, completing a program for such groups started over a decade ago.

### Kate Petersen (UMN Duluth)

*PSL(2,C) representations of knot groups*

I will discuss a method of producing defining equations for representation varieties of the canonical component of a knot group into PSL2(C). This method uses only a knot diagram satisfying a mild restriction and is based upon the underlying geometry of the knot complement. In particular, it does not involve any polyhedral decomposition or triangulation of the link complement. This is joint work with Anastasiia Tsvietkova.

### Homin Lee (Indiana)

*Smooth actions on manifolds by higher rank lattices*

We will discuss the rigidity phenomenon of smooth actions on compact manifolds by higher rank lattices such as [math]\displaystyle{ SL(3,Z) }[/math] or [math]\displaystyle{ SL(2, Z[\sqrt{17}]) }[/math] following the philosophy of Zimmer program. Especially, we consider the case when action posses a hyperbolic phenomenon. In this talk, we discuss 1) Anosov actions on n manifolds by lattices in SL(n,R), and 2) nonuniform hyperbolic actions on n manifolds by lattices in [math]\displaystyle{ SL(n,R) }[/math] (ongoing work with Aaron Brown). If time permit, we also discuss about Anosov actions by higher rank lattices without Property (T).

### Sunrose Shrestha (Wesleyan)

*Periodic directions on the Mucube*

The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edgeto- edge via translations) has been widely studied due to connections to polygonal billiards and Teichmüller theory. However, much less is known regarding straight-line flows on non-compact infinite translation surfaces. In this talk we will review work on straight line flows on infinite translation surfaces and consider such a flow on (the translation cover of) the Mucube– an infinite Z^3 periodic half-translation square-tiled surface– first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions on the Mucube– in terms of a subgroup of SL(2,Z). This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.

### Sam Taylor (Temple)

*Endperioidic maps via pseudo-Anosov flows*

We show that every atoroidal, endperiodic map of an infinite-type surface is isotopic to a homeomorphism that is naturally the first return map of a pseudo-Anosov suspension flow on a fibered manifold. Morally, these maps are all obtained by “spinning” fibers around transverse surfaces in the boundary of the fibered face. The structure associated to these spun pseudo-Anosov maps allows for several applications. These include defining and charactering stretch factors of endperiodic maps, relating Cantwell—Conlon foliation cones to Thurston’s fibered cones, and defining a convex entropy function on these cones that extends log(stretch factor).

This is joint work with Michael Landry and Yair Minsky.

## Fall 2021

## Fall Abstracts

### Nate Fisher (UW Madison)

*Boundaries, random walks, and nilpotent groups*

In this talk, we will discuss boundaries and random walks in the Heisenberg group. We will discuss a class of sub-Finsler metrics on the Heisenberg group which arise as the asymptotic cones of word metrics on the integer Heisenberg group and describe new results on the boundaries of these polygonal sub-Finsler metrics. After that, we will explore experimental work to examine the asymptotic behavior of random walks in this group. Parts of this work are joint with Sebastiano Nicolussi Golo.

### Caglar Uyanik (UW Madison)

*Dynamics on currents and applications to free group automorphisms*

Currents are measure theoretic generalizations of conjugacy classes on free groups, and play an important role in various low-dimensional geometry questions. I will talk about the dynamics of certain "generic" elements of Out(F) on the space of currents, and explain how it reflects on the algebraic structure of the group.

### Michelle Chu (UIC)

*Prescribed virtual torsion in the homology of 3-manifolds*

Hongbin Sun showed that a closed hyperbolic 3-manifold virtually contains any prescribed torsion subgroup as a direct factor in homology. In this talk we will discuss joint work with Daniel Groves generalizing Sun’s result to irreducible 3-manifolds which are not graph-manifolds.

### Osama Khalil (Utah)

*Generalized Hecke Operators and Mahler’s Problem in Diophantine Approximation*

Khintchine's Theorem provides a zero-one law describing the approximability of typical points by rational points. In 1984, Mahler asked whether the same holds for Cantor’s middle thirds set. His question fits into a long studied line of research aiming at showing that Diophantine sets are highly random and are thus disjoint, in a suitable sense, from highly structured sets.

We will discuss the first complete analogue of Khintchine’s theorem for certain self-similar fractal measures, recently obtained in joint work with Manuel Luethi. The key ingredient in the proof is an effective equidistribution theorem for fractal measures on the space of unimodular lattices, generalizing a long history of similar results for smooth measures beginning with Sarnak’s work in the eighties. To prove the latter, we associate to such fractals certain p-adic Markov operators, reminiscent of the classical Hecke operators, and leverage their spectral properties. No background in homogeneous dynamics will be assumed.

### Theodore Weisman (UT Austin)

*Relative Anosov representations and convex projective structures*

Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.

In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.

### Grace Work (UW Madison)

*Parametrizing transversals to horocycle flow*

There are many interesting dynamical flows that arise in the context of translation surfaces, including the horocycle flow. One application of the horocycle flow is to compute the distribution of the gaps between slopes of saddle connections on a specific translation surface. This method was first developed by Athreya and Chueng in the case of the torus, where the question can be restated in terms of Farey fractions and was solved by R. R. Hall using methods from analytic number theory. An important step in this process is to find a good parametrization of a transversal to horocycle flow. We will show how to do this explicitly in the case of the octagon, how it generalizes to a specific class of translation surfaces, lattice surfaces, (both joint work with Caglar Uyanik), and examine how to parametrize the transversal for a generic surface in a given moduli space.

### Chenxi Wu (UW Madison)

The Hubbard tree is a combinatorial object that encodes the dynamic of a post critically finite polynomial map, and its topological entropy is called the core entropy. I will talk about an upcoming paper with Kathryn Lindsey and Giulio Tiozzo where we provide geometric constrains to the Galois conjugates of exponents of core entropy, which gives a necessary condition for a number to be the core entropy for a super attracting parameter.

### Jack Burkart (UW Madison)

*Geometry and Topology of Wandering Domains in Complex Dynamics*

Let f: C --> C be an entire function. In complex dynamics, the main objects of study are the Fatou set, the points where f and its iterates locally form a normal family, and the Julia set, which is the complement of the Fatou set and often has a fractal structure. The Fatou set is open, and connected components of the Fatou set map to each other. Connected components that are not periodic are called wandering domains.

In this talk, I give a biased survey of what we know about the existence of wandering domains in complex dynamics and their geometry and topology, highlighting both classical and recent results and some open problems.

### Jayadev Athreya (UW Seattle)

*Stable Random Fields, Patterson-Sullivan Measures, and Extremal Cocycle Growth*

We study extreme values of group-indexed stable random fields for discrete groups G acting geometrically on spaces X in the following cases: (1) G acts freely, properly discontinuously by isometries on a CAT(-1) space X, (2) G is a lattice in a higher rank Lie group, acting on a symmetric space X, (3) G is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group G on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth which measures the distortion of measures on the boundary in comparison to the movement of points in the space X and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle U(X/G) provided X/G has non-arithmetic length spectrum. This is joint work with Mahan MJ and Parthanil Roy.

### Funda Gültepe (U Toledo)

*A universal Cannon-Thurston map for the surviving curve complex*
Using the Birman exact sequence for pure mapping class groups, we construct a universal Cannon-Thurston map onto the boundary of a curve complex for a surface with punctures we call surviving curve complex. In this talk, I will give some background and motivation for the work and I will explain the ingredients of the construction and then give a sketch of proof of the main theorem. Joint work with Chris Leininger and Witsarut Pho-on.

### Jonah Gaster (UWM)

*The Markov ordering of the rationals*

A rational number p/q determines a simple closed curve on a once-punctured torus, which then has a well-defined length when the torus is endowed with a complete hyperbolic metric. When the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance, with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the (q,p)-plane.

### Chloe Avery (U Chicago)

*Stable Torsion Length*

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples. This is joint work with Lvzhou Chen.

### Daniel Levitin (UW Madison)

*Metric Spaces of Arbitrary Finitely-Generated Scaling Group*

A quasi-isometry between uniformly discrete spaces metric spaces of bounded geometry is scaling if it is coarsely k-to-1 for some positive real k, up to some error that takes into account the geometry of the space. The collection of k for which scaling self-maps exist is a multiplicative group by composing maps. Scaling maps and the scaling group have been used to prove a variety of quasi-isometric rigidity theorems for groups and spaces. In this talk, I construct a space with any finitely-generated scaling group.

## Archive of past Dynamics seminars

2020-2021 Dynamics_Seminar_2020-2021