Representations of surface groups
March 19 to March 23, 2007
at the
American Institute of Mathematics,
Palo Alto, California
organized by
Steven Bradlow,
Oscar GarciaPrada,
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
multicomponent 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 algebrogeometric 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:
 split $\mathbb{R}$forms of semisimple groups;
 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 Zariskidense 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:
 Relation of the different Lagrangian subbundles constructed using the
harmonic map (in the Higgs bundle approach) and the boundary map;

Coordinates and
parametrizations of the Hitchin component and of the space of maximal
representations.
 Structures on the moduli space: symplectic, holomorphic,
Kahler;

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
FockGoncharov coordinates
Hecke correspondence
Narasimhan Seshadri theorem discussion group.
Reading material
The organizers have written some introductory
material. (45 page PDF)
Introductory material on Hermitian symmetric spaces by Tobias Strubel.