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)	Comparing length functions for actions on CAT(0) cube complexes	Levitin,Uyanik
October 14	Francis Bonahon (USC/Michigan State)	The symplectic structure of the Hitchin component	Loving
October 21	Dongryul Kim (Yale)	Conformal measure rigidity and ergodicity of horospherical foliations	Uyanik
October 28	Matthew Durham (UC Riverside)	Cubical metrics on mapping class groups, Z-boundaries, and the Farrell--Jones Conjecture	Loving
November 4	Caglar Uyanik (UW)	Cannon-Thurston maps, random walks, and rigidity	local
November 11	Fernando Al Assal (UW)	Asymptotically geodesic surfaces in hyperbolic 3-manifolds	local
November 18	Paige Hillen (UCSB)	Latent symmetry of graphs and stretch factors in Out(Fn)	Dymarz, Uyanik
November 25	Yan Mary He (Oklahoma)	Dimension theory of endomorphisms of CP^k	Wu
December 2	Alex Nolte (Rice)	Foliations in PSL(4,R)-Teichmüller theory	Zimmer
December 9

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.

Eduardo Reyes

The marked length spectrum of a closed, negatively curved Riemannian manifold records the lengths of closed geodesics. We can compare two Riemannian metrics using their marked length spectra, which can be done dynamically via the geodesic flow. This perspective has been extended to other geometric contexts, such as pairs of Anosov representations and actions on metric trees. In this talk, I will discuss a joint work with Stephen Cantrell in which we compare length functions of actions on CAT(0) cube complexes. These are non-positively curved spaces of combinatorial nature that generalize simplicial trees. The role of the geodesic flow is now played by a finite-state automaton, inspired by Calegari-Fujiwara's work about word metrics on hyperbolic groups.

Francis Bonahon

The Hitchin component of a closed surface is a preferred component in the variety of flat bundles over the surface, whose elements enjoy strong algebraic, geometric and dynamical properties.It comes with a natural symplectic structure defined by the Atiyah-Bott-Goldman form. I will provide an explicit formula expressing this symplectic form in terms of generalized Fock-Goncharov coordinates for the Hitchin component. This is joint work with Yasar Sözen and Hatice Zeybek. 

Dongryul Kim

As generalizations of Mostow's rigidity theorem, Sullivan, Tukia, Yue, and Oh and I proved rigidity theorems for representations of rank-one discrete subgroups of divergence type, in terms of the push-forwards of conformal measures via boundary maps.

In this talk, I will discuss a further extension of them to a certain class of higher-rank discrete subgroups, which we call hypertransverse subgroups. This class includes all rank one discrete subgroups, Anosov subgroups, relatively Anosov subgroups, and notably, their subgroups. The proof is developing the idea of the joint work with Oh for self-joinings of higher-rank hypertransverse subgroups, overcoming the lack of CAT(-1) geometry on symmetric spaces. In contrast to the work of Sullivan, Tukia, and Yue, the argument is closely related to studying the ergodicity of horospherical foliations.

Matthew Durham

Our understanding of the geometry of mapping class groups of finite-type surfaces has seen a number of leaps forward in recent years with the influx of techniques from CAT(0) cube complexes, including the cubical approximation techniques of Behrstock--Hagen--Sisto and the wallspace constructions of Petyt--Zalloum.

In this talk, I'll explain work with Yair Minsky and Alessandro Sisto in which we prove that mapping class groups are coherently locally modeled by cube complexes like an atlas of charts on a manifold. This allows us to build many new metrics on mapping class groups which closely behave like metrics on CAT(0) cube complexes.

Our main application is the construction of a geometric action of the mapping class group on asymptotically CAT(0) space (in the sense of Kar). This allows us to construct the first construction of a Z-boundary for mapping class groups (in the sense of Bestvina) and give a new proof of the Farrell--Jones Conjecture for them, originally due to Bartels--Bestvina.

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.

Fernando Al Assal

Let M be a hyperbolic 3-manifold. We say a sequence of distinct (non-commensurable) essential closed surfaces in M is asymptotically geodesic if their principal curvatures go uniformly to zero. When M is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they might not equidistribute. We will also talk about the fact that such sequences do not exist when M is geometrically finite of infinite volume. Finally, time permitting, we will discuss some partial answers to the question: does the existence of asymptotically geodesic surfaces in a negatively-curved 3-manifold imply the manifold is hyperbolic? This joint work with Ben Lowe.

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.

Yan Mary He

Let k \ge 1 be an integer and let f: CP^k \to CP^k be a holomorphic endomorphism of algebraic degree at least 2. Let \nu be an invariant ergodic probability measure with positive Lyapunov exponents. In this talk, we will introduce a volume dimension of the measure \nu which is equivalent to the Hausdorff dimension when k =1 but depends on the dynamics of the map when k \ge 2 to incorporate the non-conformality of holomorphic maps in higher dimensional projective spaces. We prove a generalized Mane-Manning formula relating the dimension, entropy and Lyapunov exponents of \nu. As applications we will characterize the first zero of a pressure function for expanding invariant measure in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. The talk is based on joint work with Fabrizio Bianchi.

Alex Nolte

We will present a classification of group-invariant foliations of a domain of discontinuity for PSL(4,R)-Hitchin representations whose leaves are properly convex domains in projective spaces. Two foliations of this form are the starting point of Guichard-Wienhard’s beautiful framework to describe the projective geometry of PSL(4,R)-Hitchin representations; the main result is that there is exactly one other similar foliation. We will emphasize how the special structure of Hitchin representations can be used to understand the shape of these domains of discontinuity, and the role this plays in the proof.

Spring 2025

date	speaker	title	host(s)
January 27	Ben Stucky (Beloit)	TBA	semi-local
February 10	Nate Fisher (Swarthmore)	TBA	Dymarz
March 3	Nick Salter (Notre Dame)	TBA	Apisa
March 17	Yongquan Zhang (Stony Brook)	TBA	Kent
April 14	Nick Miller (OU)	TBA	Loving and Uyanik
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