# 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? | Uyanik and Zimmer |

September 16 | Ben Lowe (Chicago) | Rigidity and Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature | Al Assal |

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) | TBA | Uyanik |

October 7 | Eduardo Reyes (Yale) | TBA | Levitin and 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) | TBA | Dymarz |

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

### Eduardo Reyes

### Francis Bonahon

### Dongryul Kim

### 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

## Spring 2025

date | speaker | title | host(s) |
---|---|---|---|

January 27 | Ben Stucky (Beloit) | TBA | semi-local |

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