Group Actions and Dynamics Seminar

During the Fall 2024 semester, RTG / Group Actions and Dynamics seminar meets in room B325 Van Vleck 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 2024

date	speaker	title	host(s)
September 9	Max Lahn (Michigan)	Which Reducible Representations are Anosov?	Loving,Uyanik,Zimmer
September 16	Ben Lowe (Chicago)	Rigidity and Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature	Al Assal, Uyanik
September 23	Harrison Bray (George Mason)	A 0-1 law for horoball packings of coarsely hyperbolic metric spaces and applications to cusp excursion	Zimmer
September 30	Eliot Bongiovanni (Rice)	Extensions of finitely generated Veech groups	Uyanik
October 7	Eduardo Reyes (Yale)	TBA	Levitin,Uyanik
October 14	Francis Bonahon (USC/Michigan State)	TBA	Loving
October 21	Dongryul Kim (Yale)	TBA	Uyanik
October 28	Matthew Durham (UC Riverside)	TBA	Loving
November 4	Caglar Uyanik (UW)	Cannon-Thurston maps, random walks, and rigidity	local
November 11	Fernando Al Assal (UW)	TBA	local
November 18	Paige Hillen (UCSB)	Latent symmetry of graphs and stretch factors in Out(Fn)	Dymarz, Uyanik
November 25	Thanksgiving week
December 2	reserved	TBA	TBA
December 9	reserved	TBA	TBA

Fall Abstracts

Max Lahn

We present a characterization of the Anosov condition adapted for linear representations which preserve a flag. Key to this characterization is the consistency of large eigenvalue configurations, which we will motivate and explore in detail. We'll see that this line of reasoning gives rise to a convex deformation space of reducible Anosov representations, and conclude that for many word hyperbolic groups, any connected component of the character variety consisting of Anosov representations cannot contain reducible representations. Time permitting, we'll discuss work in progress to further characterize the convex domains of Anosov representations.

Ben Lowe

This talk will discuss progress towards answering the following question: if a closed negatively curved manifold M has infinitely many totally geodesic hypersurfaces, then must it have constant curvature (and thus, by work of Bader-Fisher-Stover-Miller and Margulis-Mohamaddi, be homothetic to an arithmetic hyperbolic manifold)? First, I will talk about work that gives a partial answer to this question under the assumption that M is hyperbolizable. This part uses Ratner’s theorems in an essential way. I will also talk about work that gives an affirmative answer to the question if the metric on M is analytic. In this case the analyticity condition together with properties of the geodesic flow in negative curvature allow us to conclude what in constant curvature would follow from Ratner’s theorems. This talk is based on joint work with Fernando Al Assal, and Simion Filip and David Fisher.

Harrison Bray

On the cusp of the 100 year anniversary, Khinchin's theorem implies a strong 0-1 law for the real line; namely, the set of real numbers within distance q^{-2-\epsilon} of infinitely many rationals p/q is Lebesgue measure 0 for \epsilon>0, and full measure for \epsilon=0. In these lectures, I will present an analogous result for horoball packings in Gromov hyperbolic metric spaces. As an application, we prove a logarithm law; that is, we prove asymptotics for the depth in the packing of a typical geodesic. This is joint work with Giulio Tiozzo.

Eliot Bongiovanni

Given a closed surface S, a subgroup G of the mapping class group of S has an associated extension group Γ, which is the fundamental group of an S-bundle. A general problem is to infer features of Γ from G: I take G to be a finitely generated Veech group and show that Γ is hierarchically hyperbolic. This is a generalization of results from Dowdall, Durham, Leininger, and Sisto regarding lattice Veech groups. After a quick primer on flat surfaces and Veech groups, the focus of this talk is constructing a hyperbolic space Ê on which Γ acts nicely (isometrically and cocompactly). I will also discuss how this example contributes to the growing evidence of a good notion of “geometric finiteness” for subgroups of mapping class groups.

Caglar Uyanik

Cannon and Thurston showed that a hyperbolic 3-manifold that fibers over the circle gives rise to a sphere-filling curve. The universal cover of the fiber surface is quasi-isometric to the hyperbolic plane, whose boundary is a circle, and the universal cover of the 3-manifold is 3-dimensional hyperbolic space, whose boundary is the 2-sphere. Cannon and Thurston showed that the inclusion map between the universal covers extends to a continuous map between their boundaries, whose image is onto. In particular, any measure on the circle pushes forward to a measure on the 2-sphere using this map. We compare several natural measures coming from this construction.

Paige Hillen

Given an irreducible element of Out(Fn), there is a graph and an irreducible "train track map" on this graph, which induces the outer automorphism on the fundamental group. The stretch factor of an outer automorphism measures the asymptotic growth rate of words in Fn under applications of the automorphism, and appears as the leading eigenvalue of the transition matrix of such a train track representative. I'll present work showing a lower bound for the stretch factor in terms of the edges in the graph and the number of folds in the fold decomposition of the train track map. Moreover, in certain cases, a notion of the latent symmetry of a graph G gives a lower bound on the number of folds required for any irreducible train track map on G. I'll use this to classify all single fold train track maps.

Spring 2025

date	speaker	title	host(s)
January 27	Ben Stucky (Beloit)	TBA	semi-local
February 10	Nate Fisher (Swarthmore)	TBA	Dymarz
April 21	Mladen Bestvina (Utah)	Distinguished Lecture Series	Uyanik
April 28	Inanc Baykur (UMass)	TBA	Uyanik

Archive of past Dynamics seminars

2023-2024 Dynamics_Seminar_2023-2024

2022-2023 Dynamics_Seminar_2022-2023

2021-2022 Dynamics_Seminar_2021-2022

2020-2021 Dynamics_Seminar_2020-2021