Dynamics Seminar

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During the Fall 2023 semester, RTG / Group Actions and Dynamics seminar meets in room Sterling Hall 1339 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.

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.

Spring 2023

date speaker title host(s)
January 29 Michael Zshornack (UCSB) TBA Zimmer
February 5 Sayantan Khan (Michigan) TBA Uyanik
February 19 Yulan Qing (Tennessee) TBA Zimmer
April 29 Chris Leininger (Rice) TBA Uyanik

Archive of past Dynamics seminars

2022-2023 Dynamics_Seminar_2022-2023

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021