## FRG Lecture Series

## Geometric Structures on Higher Teichmüller Spaces

## December 4-7, 2019

## University of Michigan, Ann Arbor

### Sponsored by NSF grant DMS-1564362

## Minicourse

### December 4-6

### Teichmüller theory on moduli spaces of Anosov flow (Lecture notes)

### Nicolas Tholozan (ENS)

## Lecture Series

### December 6-7

### Sam Ballas (FSU)

### Jean-Philippe Burelle (Sherbrooke)

### Brian Collier (Riverside)

### Xian Dai (Rice)

### Xuesen Na (Maryland)

### Andrea Tamburelli (Rice)

### Anibal Velozo (Yale)

### Feng Zhu (Michigan)

Organizers:

Martin Bridgeman (Boston College)

Dick Canary (University of Michigan)

Richard Wentworth (University of Maryland)

Michael Wolf (Rice University)

### Sponsored by NSF grant DMS-1564362

## FRG Lecture Series Schedule

### Wednesday December 4

4:00–5:30pm **Nicolas Tholozan: Teichmüller theory on moduli spaces of Anosov flows I: Reparametrizations of the geodesic flow, **170 Weiser Hall

### Thursday December 5

3:00–4:00pm **Open discussion,** 3866 East Hall

4:00–5:30pm: **Nicolas Tholozan: Teichmüller theory on moduli spaces of Anosov flows II: Diffeomorphisms of the circle,** 1360 East Hall

### Friday December 6

1:00–1:45pm: **Brian Collier: A general Cayley correspondence for higher Teichmüller spaces,** 2306 Mason Hall

2:00–2:45pm: **Xuesen Na: Limiting Configuration and Spectral Data of SU(1,2) Higgs Bundle,** 2306 Mason Hall

3:00–4:00pm **Open discussion,** 3866 East Hall

4:00–5:30pm: **Nicolas Tholozan: Teichmüller theory on moduli spaces of Anosov flows III: Teichmüller space of the stable foliation,** B844 East Hall

### Saturday December 7

1360 East Hall

9:15–10:00am **Jean-Philippe Burelle: Schottky groups in higher Teichmüller theory**

10:15–11:00am **Anibal Velozo Some aspects of the thermodynamic formalism of the geodesic flow in the non-compact case**

11:30am–12:15pm **Xian Dai: Geodesic coordinates for the pressure metric along Fuchsian locus**

2:00-2:45pm **Sam Ballas: Gluing equations for projective structures on 3-manifolds**

3:00–3:45pm **Feng Zhu: Relatively dominated representations**

4:15–5:00pm **Andrea Tamburelli: Geometric structures parametrized by polynomial holomorphic differentials**

## Abstracts

### Minicourse

**Nicholas Tholozan: Teichmüller theory on moduli spaces of Anosov flows**

In the 90s, Sullivan introduced the idea of studying families of dynamical systems with hyperbolic properties as Teichmüller spaces of 2-dimensional foliations or laminations. In these lectures, we will develop this idea in the case of the family of reparametrizations of the geodesic flow of a hyperbolic surface, providing a framework in which we hope to understand better the geometry of higher Teichmüller spaces (such as Hitchin components).

**Lecture 1: Reparametrizations of the geodesic flow**

In the first lecture, we introduce the space of reparametrizations of the geodesic flow of a hyperbolic surface and describe its natural geometry.

**Lecture 2: Diffeomorphisms of the circle**

In the second lecture introduce a space of Anosov representations of a surface group into the group of diffeomorphisms of the circle, which can be viewed (to some extent) as an infinite dimensional space of maximal representations. We construct a map from this space to the space of reparametrizations and explain how to (almost) construct an inverse map, following Marguliss construction of the invariant measure of maximal entropy.

**Lecture 3: Teichmüller space of the stable foliation**

Following Sullivan, we finally introduce the Teichmüller space of the weakly stable foliation of the geodesic flow. We explain why it is (more or less) in bijection with the space of reparametrizations. If time allows, we will discuss preliminary results on the relation between the pressure and WeilPetersson metrics.

### Lecture Series

**Sam Ballas: Gluing equations for projective structures on 3-manifolds**

In his seminal notes, Thurston describes how one can associate to a triangulated 3-manifold, M, a system of complex valued equations and inequalities whose solutions encode hyperbolic structures M. In this talk we describe a recent generalization in which we again start with a triangulated 3-manifold M and construct a system of real valued equations and inequalities whose solutions encode real projective structures on M. Furthermore, there are several geometrically interesting projective structures (i.e. hyperbolic, anti de Sitter, properly convex) that arise from this construction. This work is joint with A. Casella.

**Jean-Philippe Burelle: Schottky groups in higher Teichmüller theory**

A Fuchsian Schottky group is a free subgroup of hyperbolic isometries with simple prescribed dynamics on the hyperbolic plane. The Teichm ̈uller space of a surface with non-empty boundary can be identified with the space of Schottky groups with a given combinatorial structure. I will explain how various types of higher Teichm ̈uller spaces can be similarly described with Schottky groups, how more general Anosov Schottky groups can be defined using geometric objects and what can be said about their moduli spaces. This talk will include parts of joint works with N. Treib, F. Kassel, and V. Charette.

**Brian Collier: A general Cayley correspondence for higher Teichmüller spaces**

In this talk I will present a Higgs bundle parameterization of all expected higher Teichmüller spaces. This construction depends on special sl2 subalgebras and generalizes Hitchin description of the Hitchin component. The relationship with Guichard-Wienhard’s notion of positive Anosov representations will also be discussed.

**Xian Dai: Geodesic coordinates for the pressure metric along Fuchsian locus.**

I will discuss my recent result that rays parameterized by quadratic and cubic differentials in Hitchin base for Hitchin component of PSL(3, R) case yield geodesic coordinates at any Fuchsian point with respect to pressure metric. The computation is a combination of thermodynamic formalism and Higgs bundle technique. We study flat connections from Hitchin’s equations and their parallel transports to obtain variations of reparametrization functions for the pressure metric.

**Xuesen Na: Limiting Configuration and Spectral Data of SU(1,2) Higgs Bundle**

In this talk I will introduce the notion of limiting configuration of Hitchin equation invented by Mazzeo et al (2014) and on-going work to extend it to study generic ends of moduli space of SU(1,2) Higgs bundle. I will also present an explicit description of spectral data of SU(1,2) Higgs bundle.

**Andrea Tamburelli: Geometric structures parametrized by polynomial holomorphic differentials**

The work of Tam et al. on harmonic maps from the complex plane to the hyperbolic plane and the work of Dumas-Wolf on planar affine spheres suggest that polynomial holomorphic differentials should parametrize geometric structures with some combinatorial flavor. In this talk we will explain how to construct light- like polygons in the 2- and 3-dimensional Einstein Universe starting from quadratic and quartic polynomial differentials on the complex plane.

**Anibal Velozo: Some aspects of the thermodynamic formalism of the geodesic flow in the non-compact case**

Thermodynamic formalism is a subject in dynamical systems that studies quantities such as the en- tropy/pressure and has been very useful to understand hyperbolic dynamical systems, which are chaotic. In this talk I plan to discuss the thermodynamic formalism of the geodesic flow on a non-compact pinched negatively curved manifold, with emphasis on the differences to the compact case (which is much better understood). The connection between thermodynamic formalism and equidistribution/counting of closed geodesics was noticed by Margulis and Bowen in the 70’s and will play an important role in our discussion. If time permits, analogies with symbolic dynamics will be presented.

**Feng Zhu: Relatively dominated representations**

Convex cocompact subgroups of rank-one semisimple Lie groups such as PSL(2,R) form a structurally stable class of quasi-isometrically embedded discrete subgroups which are naturally associated to negatively- curved geometric structures. Anosov representations give a higher-rank analogue of convex cocompactness which shares many of its good geometric and dynamical properties, and have become important objects of study in higher Teichmueller theory. I will introduce a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness—a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity.

## Participants

Caleb Ashley (Michigan)

Sam Ballas (Florida State)

Martin Bobb (UT Austin)

Harry Bray (Michigan)

Martin Bridgeman (Boston College)

JP Burelle (Sherbrooke)

Karen Butt (Michigan)

Richard Canary (University of Michigan)

Brian Collier (UC-Riverside)

Tomasso Cremaschi (USC)

Xian Dai (Rice)

Didac Martinex Granado (Indiana)

Mark Greenfield (Michigan)

Zeno Huang (CUNY)

Mitul Islam (Michigan)

Caleb Dilsavor (Ohio State)

Giuseppe Martone (Michigan)

Xuesen Na (Maryland)

Thang Nguyen (Michigan)

Charles Ouyang (Rice)

Franco Vargas Pallette (Yale)

Samantha Pinella (Michigan)

Max Riestenberg (UT Austin)

Salman Siddiqi (Michigan)

Ralf Spatzier (Michigan)

Andrea Tamburelli (Rice)

Nicolas Tholozan (ENS-Paris)

Dan Thompson (Ohio State)

Kostas Tsouvalas (Michigan)

Anibal Velozo (Yale)

Richard Wentworth (University of Maryland)

Michael Wolf (Rice University)

Scott Wolpert (Maryland)

Alex Wright (Michigan)

Feng Zhu (Michigan)

Bradley Zykowski (Michigan)