# Representations of surface groups

March 19 to March 23, 2007

at the

American Institute of Mathematics, San Jose, California

organized by

Steven Bradlow, Oscar Garcia-Prada, William M. Goldman, and Anna Wienhard

## Original Announcement

This workshop will bring together researchers studying representations of fundamental groups of Riemann surfaces into real semsimple Lie groups. Such representations form multi-component algebraic sets. Recent progress in understanding these components has come from quite different approaches:
• Higgs bundle techniques, which combine (twisted) harmonic maps into symmetric spaces and holomorphic vector bundles; Morse theory methods on the moduli spaces of Higgs bundles have led to the calculation of the cohomology of these components;
• The bounded cohomological, the dynamical and the algebro-geometric approach relate the geometry of flag varieties at the boundary of symmetric spaces to the dynamics of surface group actions. This has led to a geometric understanding of surface group actions in certain components of the moduli space of representations.
The main goal of the workshop is to clarify the relations between these different approaches to initiate further research in this area. Two classes of Lie groups have so far been studied, namely:
1. split $\mathbb{R}$-forms of semisimple groups;
2. automorphism groups of Hermitian symmetric spaces.
In both cases, special components are distinguished by notions of positivity in the flag variety. For split groups, Hitchin found components containing Teichmuller space which are homemorphic to cells, whereas for groups of Hermitian type, there are components of maximal characteristic number (Toledo invariant) comprising certain equivalence classes of discrete Zariski-dense embeddings. Higgs bundle methods have successfully computed the homotopy type of the moduli spaces, whereas the other approaches have successfully identified the dynamical property of the corresponding representations.

A central focus of this workshop will be the real symplectic group $Sp(2n,\mathbb{R})$, which is the unique $\mathbb{R}$-split Lie group of Hermitian type. Specific topics to be addressed include:

1. Relation of the different Lagrangian subbundles constructed using the harmonic map (in the Higgs bundle approach) and the boundary map;
2. Coordinates and parametrizations of the Hitchin component and of the space of maximal representations.
3. Structures on the moduli space: symplectic, holomorphic, Kahler;
4. Mapping class group action and the energy function on Teichmuller space.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Group photo(1.7 Meg) or another group photo(1.7 Meg)

## Wiki

Graeme Wilkin has started a surface groups wiki. (password protected. login "surface", pword "groups").

## Notes from the workshop

Teichmuller theory
Introduction to harmonic mappings
Maximal representations by Burger
Parabolic Higgs bundle and representations by Buquard
Lifting representations to the mapping class group by Kotschick
Bounded cohomology
Energy functional discussion
Fock-Goncharov coordinates
Hecke correspondence
Narasimhan Seshadri theorem discussion group.