Dynamics Seminar 2023-2024

During the Spring 2024 semester, RTG / Group Actions and Dynamics seminar meets in room Sterling Hall 3425 on Mondays from 2:25pm - 3:15pm. 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 2024

date	speaker	title	host(s)
January 22	Aaron Messerla (UIC)	Quasi-isometries of relatively hyperbolic groups with an elementary hierarchy	Loving and Uyanik
January 24 (1pm in VV 901)	Mitul Islam (Max-Planck-Institut)	Morse-ness in convex projective geometry	Zimmer
January 29	Michael Zshornack (UCSB)	Rational surface groups on the Hitchin component	Zimmer
February 5	Sayantan Khan (Michigan)	Bootstrapping dynamics in the moduli space of non-orientable surfaces	Uyanik
February 12	Noelle Sawyer (Southwestern)	Unique Equilibrium States for Geodesic Flows	Loving, Uyanik, Work
February 19	Yulan Qing (Tennessee)	Geometric Boundary of Groups	Zimmer
February 26	Blandine Galiay (IHES)	Divisible convex sets in flag manifolds and rigidity	Zimmer
March 4	David Aulicino (Brooklyn College)	Siegel-Veech Constants of Cyclic Covers of Generic Translation Surfaces	Apisa
March 11	Aaron Calderon (Chicago)	Equidistribution of twist tori	Loving and Uyanik
March 18	Josh Southerland (Indiana)	Shrinking targets on square-tiled surfaces	Fisher
April 8	Matt Bainbridge (Indiana)	Moduli of affine surfaces and holomorphic connections	Apisa
April 15	Ilya Kapovich (CUNY)	Counting closed geodesics in the moduli space of Outer space: double exponential growth	Uyanik
April 22	Yu-Chan Chang (Wesleyan)	TBA	Dymarz
April 29	Chris Leininger (Rice)	Coarse geometry of surface bundles over Teichmüller curves	Kent, Loving, Uyanik
September 23	Harrison Bray (George Mason)	TBA	Zimmer

Spring Abstracts

Aaron Messerla

Sela introduced limit groups in his work on the Tarski problem, and showed that each limit group has a cyclic hierarchy. We prove that a class of relatively hyperbolic groups, equipped with a hierarchy similar to the one for limit groups, is closed under quasi-isometry. Additionally, these groups share some of the algebraic properties of limit groups. In this talk I plan to present motivation for and introduce the class of groups studied, as well as present some of the results for this class.

Mitul Islam

The (Hilbert metric) geometry of properly convex domains generalizes real hyperbolic geometry. This generalization is far from the Riemannian notion of non-positive curvature, but they have some intriguing similarities. I will explore this connection from a coarse geometry viewpoint. The focus will be on Morse geodesics ("negatively curved directions", in a coarse sense) in properly convex domains. I will show that Morse-ness can be characterized entirely using linear algebraic data (i.e. singular values of matrices that track the geodesic). Further, I will discuss how this coarse geometric notion of Morse is related to the symmetric space geometry as well as the smoothness of boundary points. This is joint work with Theodore Weisman.

Michael Zshornack

Margulis's work on lattices in higher-rank and a number of questions on the existence of surface subgroups motivate the need for understanding arithmetic properties of spaces of surface group representations. In recent work with Jacques Audibert, we outline one possible approach towards understanding such properties for the Hitchin component, one particularly nice space of representations. We utilize the underlying geometry of this space to reduce questions about its arithmetic to questions about the arithmetic of certain algebraic groups, which in turn, allows us to characterize the rational points on these components. In this talk, I'll give an overview of the geometric methods behind the proof of our result and indicate some natural questions about the nature of the resulting surface group actions that follow.

Sayantan Khan

The moduli spaces of non-orientable hyperbolic surfaces behave significantly differently from their orientable counterparts. They have infinite volume, almost all geodesic flow orbits escape off to infinity, and the growth of mapping class group orbit points and simple closed curves is believed to have non-integer exponents, unlike in the orientable setting. In this talk, we outline some of the oddities of these moduli spaces, and outline an approach for studying the dynamics on these spaces via Patterson-Sullivan theory. A key obstruction to imitating the techniques from the orientable setting is that a number of these techniques rely on the dynamics of the geodesic flow and the mapping class group action on Teichmüller spaces of orientable surfaces, which is not something we can do in the non-orientable setting, since that is what we are trying to prove. The way around these obstructions is by proving weaker versions of these dynamical statements using a dynamics free approach, which we then use to bootstrap the stronger results.

Noelle Sawyer

In this talk I will discuss some known results about the geodesics and equilibrium states of the geodesic flow in negative curvature. After, I will introduce some of the tools and techniques needed to show the uniqueness of equilibrium states in the setting of translation surfaces. If time allows, I will talk about some of our upcoming work about the Bernoulli property. This is joint work with Benjamin Call, Dave Constantine, Alena Erchenko, and Grace Work.

Yulan Qing

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this talk we generalize the Gromov boundary. We first construct the sublinearly Morse boundaries and show that it is a QI-invariant, metrizable topological space. We show sublinearly Morse directions are generic both in the sense of Patterson-Sullivan and in the sense of random walk.

The sublinearly Morse boundaries are subsets of all directions with desired properties. In the second half of the talk, we will truly consider the space of all directions and show that with some minimal assumptions on the space, the resulting boundary is a QI-invariant topology space in which many existing boundaries are embedded. This talk is based on a series of work with Kasra Rafi and Giulio Tiozzo.

Blandine Galiay

Divisible convex sets have been widely studied since the 1960s. They are proper domains of the projective space that admit a cocompact action of a discrete subgroup of the linear projective group. The best-known examples are symmetric spaces embedded in the projective space, but there are also many nonsymmetric examples. It is natural to seek to generalize this theory, by replacing the projective space by a flag variety G/P, where G is a real semisimple non-compact Lie group and P a parabolic of G. A question of van Limbeek and Zimmer is then: are there examples of divisible convex sets in G/P that are nonsymmetric? In a number of cases, it has been proved that there are not. In this talk, we will focus on some particular classes of flag varieties in which rigidity can indeed be observed.

David Aulicino

We consider generic translation surfaces of genus g>0 with marked points and take covers branched over the marked points such that the monodromy of every element in the fundamental group lies in a cyclic group of order d. Given a translation surface, the number of cylinders with waist curve of length at most L grows like L^2. By work of Veech and Eskin-Masur, when normalizing the number of cylinders by L^2, the limit as L goes to infinity exists and the resulting number is called a Siegel-Veech constant. The same holds true if we weight the cylinders by their area. Remarkably, the Siegel-Veech constant resulting from counting cylinders weighted by area is independent of the number of branch points n. All necessary background will be given and a connection to combinatorics will be presented. This is joint work with Aaron Calderon, Carlos Matheus, Nick Salter, and Martin Schmoll.

Aaron Calderon

Given a hyperbolic surface and a collection of simple closed geodesics, one can build a family of related metrics by cutting open the surface and twisting along the geodesics. This creates to an immersed ``twist torus’’ inside the moduli space of hyperbolic structures, which turns out to be a minimal set for the unipotent-like "earthquake flow." Maryam Mirzakhani famously asked if these twist tori equidistribute when pushed forward under a corresponding geodesic(-like) flow; in this talk, I will explain joint work with James Farre in which we prove that they do equidistribute in some cases, and that they do not in others. The key tool is a bridge that allows for the transfer of ergodic-theoretic results between flat and hyperbolic geometry.

Josh Southerland

In this talk, we will study a shrinking target problem for square-tiled surfaces. A square-tiled surface is a type of translation surface which arises as a branched cover of the torus (branched over one point). The moduli space of translation surfaces carries an action of $SL^+_2(\R)$, and the stabilizer of this action is called the Veech group. We will show that the action of a subgroup of the Veech group of a square-tiled surface exhibits Diophantine-like properties. This generalizes the work of Finkelshtein, who studied a similar problem on the torus.

Matt Bainbridge

A complex affine structure on a Riemann surface is an atlas of charts (covering the complement of a discrete set of cone singularities) with complex-affine transition functions. These affine structures generalize several structures on Riemann surfaces that are of great interest in geometry and dynamics, such as translation structures, dilation structures, and flat metrics. Veech, in his paper "flat surfaces" constructed the moduli space of affine surfaces and established several foundational results.

Every affine surface has a holonomy character, which is a cohomology class recording the holonomy of the affine structure around any closed curve. This defines a holonomy map from the moduli space of affine surfaces to a torus parameterizing holonomy characters. One of Veech's main results showed that this holonomy map is a submersion away from the locus of translation surfaces.

In this talk, we'll promote a more differential (or algebraic)-geometric definition of an affine surface as a Riemann surface together with a holomorphic connection with regular singularities on its tangent bundle.  This perspective allows us to apply tools from complex algebraic geometry, in particular deformation theory, to the study of affine surfaces.  This yields more conceptual proofs of Veech's results as well as generalizations to surfaces with arbitrary cone singularities.

This talk represents joint work with Paul Apisa and Jane Wang.

Ilya Kapovich

The problem of counting closed geodesics of bounded length, originally in the setting of negatively curved manifolds, goes back to the classic work of Margulis in 1960s about the dynamics of the geodesic flow. Since then Margulis' results have been generalized to many other contexts where some whiff of hyperbolicity is present. Thus a 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi$ in the mapping class group $MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)>1$ is the ``dilatation" or ``stretch factor" of $\phi$.

We consider an analogous problem in the $Out(F_r)$ setting, for the action of the outer automorphism group $Out(F_r)$ of the free group $F_r$ of rank $r$ on a ``cousin" of the Teichmuller space, called the Culler-Vogtmann Outer space. In this context being a ``fully irreducible" element of $Out(F_r)$ serves as a natural counterpart of being pseudo-Anosov. Every fully irreducible $\phi\in Out(F_r)$ acts on the Outer space as a loxodromic isometry with translation length $\log\lambda(\phi)$, where again $\lambda(\phi)$ is the stretch factor of $\phi$. We estimate the number $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. The number $N_r(L)$ can also be interpreted as the number of homotopy classes of closed “contracting" geodesics in the moduli space of the Outer space. We prove, for $r\ge 3$, that $N_r(L)$ grows \emph{doubly exponentially} in $L$ as $L\to\infty$, in terms of both lower and upper bounds. This result reveals new behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.

Chris Leininger

A Teichmüller curve is a totally geodesically embedded hyperbolic surface in the moduli space of Riemann surfaces, and the pull-back of the "universal curve" is a naturally associated surface bundle E over this hyperbolic surface. I'll start by concretely describing E and a geometric structure on it which is a singular "hyperbolic-by-Euclidean" structure, which is a four-dimensional analogue of the singular solv metric on a fibered hyperbolic 3-manifold. I'll then talk about how to use this structure to understand the coarse geometry of its fundamental group Gamma, proving that Gamma is hierarchically hyperbolic and satisfies a strong form of quasi-isometric rigidity. This is joint work with Dowdall, Durham, and Sisto.

Fall 2023

date	speaker	title	host(s)
September 11	Vaibhav Gadre (Glasgow)	Teichmuller flow detects the fundamental group	Apisa
September 18	Becky Eastham (UW Madison)	Whitehead space: a tool to study finite regular covers of graphs	local
September 25	Brandis Whitfield (Temple)	Short curves of end-periodic mapping tori	Loving
September 28 (Thursday 4-5pm in B139)	Itamar Vigdorovich (Weizmann)	Group stability, characters, and dynamics on compact groups	Dymarz/Gurevich
October 2	Hanh Vo (Arizona State)	Short geodesics with self-intersections	Dymarz
October 9	Yandi Wu (UW Madison)	Marked Length Spectrum Rigidity for Surface Amalgams	local
October 16	Sanghoon Kwak (Utah)	Mapping class groups of Infinite graphs — “Big Out(Fn)”	Loving
October 23 (11:55-12:55 in B223)	Sara Maloni (UVA)	Dynamics on the SU(2,1)-character varieties of the one-holed torus	Uyanik
October 30	Giulio Tiozzo (Toronto)	A characterization of hyperbolic groups via contracting elements	Uyanik
November 6
November 13	Hongming Nie (Stony Brook)	A metric view of polynomial shift locus	Wu
November 20	Sam Freedman (Brown)	Periodic points of Prym eigenforms	Apisa
November 27	Luke Jeffreys (UW Madison)	Non-planarity of SL(2,Z)-orbits of origamis	local
December 4	Emily Stark (Wesleyan)	Graphically discrete groups and rigidity	Uyanik
December 11	Colloquium by Mikolaj Fraczyk at 4pm

Fall Abstracts

Vaibhav Gadre

A quadratic differential on a Riemann surface is equivalent to a half-translation structure on the surface by complex charts with half-translation transitions. The SL(2,R)-action on the complex plane takes half-translations to half-translations and so descends to moduli spaces of quadratic differentials. The diagonal part of the action is the Teichmuller flow.

Apart from its intrinsic interest, the dynamics of Teichmuller flow is central to many applications in geometry, topology and dynamics. The Konstevich—Zorich cocycle which records the action of the flow on the absolute homology of the surface, plays a key role.

In this talk, I will explain how the flow detects the topology of moduli spaces. Specifically, we will show that the flow group, namely the subgroup generated by almost flow loops, has finite index in the fundamental group. As a corollary, we will prove that the minus and plus (modular) Rauzy—Veech groups have finite index in the fundamental group, answering a question by Yoccoz.

Using this, and Filip’s results on algebraic hulls and Zariski closures of modular monodromies, we prove that the Konstevich—Zurich cocycle (separately minus and plus pieces) have a simple Lyapunov spectrum, extending the work of Forni from 2002 and Avila—Viana from 2007.

Becky Eastham

The Whitehead space of a finite regular cover of the rose is a locally infinite graph whose vertices are in one-to-one correspondence with conjugacy classes of elements of the subgroup associated with the cover. Every Whitehead space is a subgraph of the quotient of [math]\displaystyle{ \mathrm{Cay}(F_n, \mathcal{C}) }[/math] by conjugacy; here $\mathcal{C}$ is the set of elements of $F_n$ conjugate into a proper free factor. Our main interest in this space is that it is connected if and only if the fundamental group of the associated cover is generated by lifts of elements of $\mathcal{C}$ to the cover. In addition, Whitehead space of the rose is an infinite-diameter, non-hyperbolic, one-ended space with an isometric action of $\mathrm{Out}(F_n)$. Thus, Whitehead space is not quasi-isometric to the free factor complex, the free splitting complex, or Outer Space.

Brandis Whitfield

Let $S$ be a boundaryless infinite-type surface with finitely many ends and consider an end-periodic homeomorphism $f$$ $of S. The end-periodicity of $f$ ensures that $M_f$, its associated mapping torus, has a compactification as a $3$-manifold with boundary; and further, if $f$ is atoroidal, then $M_f$ admits a hyperbolic metric.

As an end-periodic analogy to work of Minsky in the finite-type setting, we show that given a subsurface $Y\subset S$, the subsurface projections between the "positive" and "negative" Handel-Miller laminations provide bounds for the geodesic length of the boundary of $Y$ as it resides in $M_f$.

In this talk, we'll discuss the motivating theory for finite-type surfaces and closed fibered hyperbolic $3$-manifolds, and how these techniques may be used in the infinite-type setting.

Itamar Vigdorovich

I will discuss three seemingly unrelated topics: 1. Stability: given a pair of matrices that almost commute, can they be perturbed to matrices which do commute? Interestingly, the answer highly depends on the chosen metric on matrices. This question is a special case of group stability: is every almost-homomorphism close to an actual homomorphism? 2. Characters: these are functions on groups with special properties that generalize the classical notion in Pontryagin's theory of abelian groups, and in Frobenius's theory of finite groups. Is every character a limit of a finite-dimensional character? 3. Topological dynamics: given a group G acting by homeomorphisms on a compact space X, are the periodic measures dense is the space all invariant measures? In this talk I will present these three subjects of study and explain how there are all in fact intimately related, as least in the amenable setting. For example, stability of the lamplighter group is strongly related to the orbit closing lemma for the Bernoulli shift, and stability of the semidirect product ZxZ[1/6] is related to whether Furstenberg's x2x3 system has dense periodic measures. The talk is based on a joint work with Arie Levit.

Hanh Vo

We consider the set of closed geodesics on a hyperbolic surface. Given any non-negative integer k, we are interested in the set of primitive essential closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.

Yandi Wu

The marked length spectrum of a negatively curved metric space can be thought of as a length assignment to every closed geodesic in the space. A celebrated result by Otal says that metrics on negatively curved closed surfaces are determined completely by their marked length spectra. In my talk, I will discuss my work towards extending Otal’s result to a large class of surface amalgams, which are natural generalizations of surfaces.

Sanghoon Kwak

Surfaces and graphs are closely related; there are many parallels between the mapping class groups of finite-type surfaces and finite graphs, where the mapping class group of a finite graph is the outer automorphism group of a free group of (finite) rank. A recent surge of interest in infinite-type surfaces and their mapping class groups begs a natural question: What is the mapping class group of an “infinite” graph? In this talk, I will explain the answer given by Algom-Kfir and Bestvina and present recent work, joint with George Domat (Rice University), and Hannah Hoganson (University of Maryland), on the coarse geometry of such groups.

Sara Maloni

In this talk we will discuss join work in progress with S. Lawton and F. Palesi on the (relative) SU(2, 1)–character variety for the once-holed torus. We consider the action of the mapping class group and describe a domain of discontinuity for this action, which strictly contains the set of convex-cocompact characters. We will also discuss the connection with the recent work of S. Schlich, and the inspiration behind this project, which lies in the rich theory developed for SL(2, C)–character varieties by Bowditch, Minsky and others.

Giulio Tiozzo

The notion of contracting element has become central in geometric group theory, singling out, in an arbitrary metric space, the geodesics which behave like geodesics in a delta-hyperbolic space. In this work, joint with K. Chawla and I. Choi, we prove the following characterization of hyperbolic groups: a group is hyperbolic if and only if the D-contracting elements are generic with respect to counting in the Cayley graph.

Hongming Nie

The escaping rates of critical points for polynomials in C[z] induce a continuous and proper map on the moduli space M_d of degree d\ge 2 polynomials. This map has a monotone-light factorization via an intermediate space T_d^* studied by DeMarco and Pilgrim. Restricting on the shift locus S_d of M_d, one obtains the corresponding intermediate space ST_d^*. In this talk, I will relate generic points in S_d to the length functions on the (2d-2)-rose graph and then present an understanding of the natural projectivization of ST_d^* from a metric view. The metric is obtained from thermodynamic metrics on the space of metric graphs. This is a joint work with Yan Mary He.

Sam Freedman

We will consider the dynamics of affine automorphisms acting on highly symmetric translation surfaces called Veech surfaces. Specifically, we’ll examine the points of the surface that are periodic, i.e., have a finite orbit under the whole automorphism group. While this set is known to be finite for primitive Veech surfaces, for applications it is desirable to determine the periodic points explicitly. In this talk we will discuss our classification of periodic points in the case of minimal Prym eigenforms, certain primitive Veech surfaces in genera 2, 3, and 4.

Luke Jeffreys

Origamis (also known as square-tiled surfaces) arise naturally in a variety of settings in low-dimensional topology. They can be thought of as generalisations of the torus (the unit square with opposite sides glued) since they are surfaces obtained by gluing the opposite sides of a collection of unit squares. There is a natural action of the matrix group SL(2,Z) on origamis. In genus two (with some extra conditions) the orbits of this action were classified by Hubert-Lelièvre and McMullen. By considering a generating set of size two for SL(2,Z) and varying the number of squares used to build the origamis, we can turn these orbits into an infinite family of four-valent graphs. It is a long-standing conjecture of McMullen that these orbit graphs form a family of expander graphs. In this talk, giving indirect evidence for this conjecture, I will discuss joint work with Carlos Matheus in which we show that these orbit graphs are eventually non-planar - a requirement of any family of expander graphs.

Emily Stark

Rigidity problems in geometric group theory frequently have the following form: if two finitely generated groups share a geometric structure, do they share algebraic structure? We consider two finitely generated groups that are either quasi-isometric or act geometrically on the same proper metric space, and we ask if they are virtually isomorphic. The work of Papasoglu--Whyte demonstrates that infinite-ended groups are quasi-isometrically flexible, but our results show that if you assume a common geometric model, then there is often rigidity. To do this, we introduce the notion of a graphically discrete group, which imposes a discreteness criterion on the automorphism group of any graph the group acts on geometrically. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; free groups are non-examples. We will present new examples and demonstrate this property is not a commensurability invariant. We will present rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Archive of past Dynamics seminars

2022-2023 Dynamics_Seminar_2022-2023

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021