**Workshop on Sp(4, ***R*)-Anosov representations

*R*)-Anosov representations

**January 10-18, 2016**

Granby, CO

Granby, CO

We will study the role of Higgs bundles, Anosov representations, Hermitian symmetric spaces and harmonic maps in maximal Sp(4,*R*) representations. After preliminary talks on symmetric spaces, Anosov representations and Higgs bundles, further topics will include: boundaries of symmetric spaces for groups of Hermitian type, domains of discontinuity, and Hitchin representations. Subsequent talks will be based on current developments in the theory. In particular, Geometric structures associated to Sp(4,*R*) Hitchin representations, Zariski closures of maximal Sp(4,*R*)representations, the Higgs bundle parameterization and connected component count of maximal Sp(4,*R*) representations and the existence of minimal surfaces associated to maximal Sp(4,*R*) representations.

The overarching theme may be summarized as follows: how can the tools of Anosov representations, Higgs bundles, Hermitian symmetric spaces and harmonic maps be used together to understand questions concerning surface group representations and the geometric structures associated to them. We hope to elucidate connections between the above techniques through a detailed of the specific case of Sp(4,*R*).

This workshop is inspired by similar successful workshops:

Workshop on Higher Teichmüller theory

Workshop on Higgs Bundles and Harmonic Maps

## Scientific Program

The workshop will consist of whiteboard talks by the participants on the following topics. Speakers will be alotted 2.5 hours per talk. Speakers will be asked to submit a 5-6 page summary, clicking on the each title below will lead you to that talks summary.

**Giuseppe Martone: Lie theory background, symmetric spaces and their boundary**

In this talk, the speaker will introduce the background material from Lie theory that will be required for the following talks. In particular, after presenting some decompositions of the semi-simple group G, the speaker will introduce (simple) roots, Weyl group, and Weyl chambers, parabolic subgroups. Then symmetric spaces will be introduced, as well as their boundaries (visual, Tits, Furstenberg). Of course, all the definition will be made concrete with examples, and with focus on the case of Sp(4,*R*).

** Suggested literature**

- Differential geometry, Lie groups and symmetric spaces; S. Helgason
- Notes from similar talk; Brian Collier notes and Tengren Zhang notes

**Fredric Palesi: Kleinian groups background**

In this talk, the speaker will introduce the notion of convex-cocompact actions on real rank one symmetric spaces. This talk will be an introduction to the following two talks about Anosov representations since these coincide with convex-cocompact actions in the real rank one case. Exemples coming from hyper bolic geometry (in dimension 2 and 3) and from complex hyperbolic geometry will be presenated.

** Suggested literature**

- Differential geometry, Lie groups and symmetric spaces; M. Kapovich

**Brice Loustau: Harmonic maps**

The speaker will recall general definitions for harmonic maps with Riemann surface domain, sketch of Wolf's harmonic map parameterization of Teichmuller space by holomorphic quadratic differentials and generalizations to higher rank, namely Corlette's theorem. The Energy functional on Teichmüller space will be defined and the fact that critical points are weakly conformal maps (equivalently branched minimal immersions) will be discussed.

**Suggested literature**

- The Teichmüller theory of harmonic maps; M. Wolf.
- Flat bundles with canonical metrics; K. Corlette
- Cross ratios, Anosov representations and the energy functional on Teichmuller space; F. Labourie
- Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature; R. Shoen, S. Yau
- Minimal immersions of closed Riemann surfaces; J. Sacks, K.Uhlenbeck

#### Daniele Alessandrini: Higgs bundles for real groups

The speaker will definition of Higgs bundles for real groups with special attention to the groups Sp(4,*R*) and maximal SL(2,*R*). A general sketch of nonabelian Hodge correspondence and Hitchin's construction of the Hitchin component (Hitchin fibration and Hitchin section) will be given with the explicit construction for Sp(4,*R*).

** Suggested literature**

- Chapter 3 of L. Schaposnik's Thesis
- Self duality equations on Riemann surfaces; N. Hitchin
- Lie groups and Teichmüller space; N. Hitchin
- Lectures by Gothen at Isaac Newton Institure for Mathematical Sciences

#### Fanny Kassel: Anosov representations via actions on boundaries and domains of discontinuity

In this talk the speaker will introduce the notion of Anosov representations, following the recent characterizations in terms of a Cartan projection of F. Gueritaud, O. Guichard, F. Kassel, A. Wienhard. In addition, the speaker will present the construction of domain of discontinuity of O. Guichard and A. Wienhard. If time permits, the talk can finish with an overview of the applications to proper actions on homogeneous spaces.

** Suggested literature**

- Anosov Representations : Domains of Discontinuity and Applications; O. Guichard, A. Wienhard
- Anosov Representations and Proper Actions; F. Gueritaud, O. Guichard, F. Kassel, A. Wienhard

#### Jeff Danciger: Anosov representations via actions on symmetric spaces and domains of discontinuity

In this talk the speaker will introduce the notion of Anosov representations, following the recent characterizations of M. Kapovich, B. Leeb, J. Porti. This approach looks at Anosov actions on the symmetric spaces, and so it generalises the well-known theory of Kleinian groups.

**Suggested literature**

- Notes from the MSRI mini-course Geometric finiteness in higher rank symmetric spaces; M. Kapovich, B. Leeb
- Dynamics at infinity of regular discrete subgroups of isometries of higher rank symmetric spaces; M. Kapovich, B. Leeb, J. Porti
- Morse actions of discrete groups on symmetric spaces; M. Kapovich, B. Leeb, J. Porti
- A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings; M. Kapovich, B. Leeb, J. Porti
- Some recent results on Anosov representations; M. Kapovich, B. Leeb, J. Porti

#### Jérémy Toulisse: Geometric structures for Hitchin representations

The speaker will explain how to identify the deformation space of convex foliated *RP*^{3}structures on the unit tangent bundle of the surface to the SL(4,*R*) Hitchin component. Special attention should be given to the additional structure induced when one specializes to the case of Sp(4,*R*) Hitchin representations instead. For Sp(4,*R*), the ideas of the Higgs bundle approach of Baraglia should be sketched.

**Suggested literature**

- Convex foliated projective structures and the Hitchin component for PSL(4,
*R*); O. Guichard, A. Wienhard - Chapter 3 of D. Baraglia's Thesis

#### Beatrice Pozzetti: Maximal representations

The speaker will explain what are maximal representations, and focus on examples pertaining to the case of Sp(4,*R*). In particular the possible Zariski closures of a maximal Sp(4,*R*) should be discussed.

**Suggested literature**

- Surface group representations with maximal Toledo invariant; M. Burger, A. Iozzi, A. Wienhard
- Maximal representations of surface groups: symplectic Anosov structures; M. Burger, A. Iozzi, F. Labourie, A. Wienhard

#### Georgios Kydonaki: Connected components of maximal $\mathsf{Sp}(4,\mathbb{R})$

The speaker will describe the connected component count of maximal Sp(4,*R*) representations and describe the Higgs bundle parmameterization of these components and possible Zariski closures of representations in each component. Special attention should be put on the 2*g* - 3 smooth components which contain only Zariski dense representations.

**Suggested literature**

- Deformations of maximal Sp(4,
*R*) surface group representations; S. Bradlow, O. Garcia-Prada, P. Gothen - Components of spaces of representations and stable triples; P. Gothen

#### Nicolaus Trieb: Toplogical invariants of Anosov representations and hybrid representations

Following the paper Topological invariants of Anosov representations, the speaker should describe how one associates invariants to connected components of Anosov representations with focus on the case of Sp(4,*R*). Also, the model representations (hybrid representations) for the 2*g* - 3 smooth components which contain only Zariski dense representations should be described.

**Suggested literature**

- Topological invariants of Anosov representations; O. Guichard, A. Wienhard

#### Andrew Sanders: Existence part of Labourie's conjecture and uniqueness for Sp(4,*R*)

Discuss Labourie's conjectured mapping class group invariant parameterization for the Hitchin component and explain how it is equivalent to the existence of a unique conformal structure in which the harmonic map to the symmetric space is a minimal immersion. Sketch how the properness of the mapping class group action and properness of the energy functional follow from maximal representations being well displacing and imply the existence of such a conformal structure for all maximal representations. Briefly describe Labourie's proof of uniqueness for Sp(4,*R*).

**Suggested literature**

- Anosov Flows, Surface Groups and Curves in Projective spaces; François Labourie
- Cross ratios, Anosov representations and the energy functional on Teichmuller space; François Labourie
- Cyclic surfaces and Hitchin components in rank 2; François Labourie

#### Jean-Philippe Burelle: Lorentzian view point of PSp(4,*R*) $\cong$ SO_{0}(2,3)-representations

Explain the low dimensional isomorphism PSp(4,*R*) $\cong$ SO_{0}(2,3) and discuss the Lorentzian geometry on the flag manifolds of the form Sp(4,*R*) / P for a maximal parabolic subgroup $\mathsf{P}\subset\mathsf{Sp}(4,\mathbb{R})$ Sp(4,*R*) / PP $\subset$ Sp(4,*R*).

**Suggested literature**

- A primer on the (2 + 1)-Einstein universe; Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, Karin Melnick

## Participants

- Daniele Alessandrini
- Jean-Philippe Burelle
- Brian Collier
- Jeff Danciger
- Sourav Ghosh
- Fanny Kassel
- Georgios Kydonakis
- Brice Loustau
- Sara Maloni
- Giuseppe Martone
- Frederic Palesi
- Beatrice Pozzetti
- Andrew Sanders
- Jérémy Toulisse
- Nicolaus Trieb

## Organizers