# 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) | 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.

### 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

### Fernando Al Assal

### 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