Dynamics Seminar

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)	TBA	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)	TBA	Dymarz
October 9	Yandi Wu (UW Madison)	TBA	local
October 16	Sanghoon Kwak (Utah)	Mapping class groups of Infinite graphs — “Big Out(Fn)”	Loving
October 23	Sara Maloni (UVA)	TBA	Uyanik
October 30	Giulio Tiozzo (Toronto)	TBA	Uyanik
November 6	Emily Stark (Wesleyan)	TBA	Uyanik
November 13	Hongming Nie (Stony Brook)	TBA	Wu
November 20	Rose Morris-Wright (Middlebury)	TBA	Dymarz
November 27	Luke Jeffreys (UW Madison)	TBA	local
December 4
December 11

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.

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.

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.

Archive of past Dynamics seminars

2022-2023 Dynamics_Seminar_2022-2023

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021