This seminar will explore dynamical systems arising from the deformation theory of representations of fundamental groups of surfaces in Lie groups. These dynamical systems arise from the actions of mapping class groups and automorphism groups of free groups on moduli spaces of flat connections over surfaces. These moduli spaces admit invariant symplectic and Poisson structures. Their Hamiltonian flows closely relate to the mapping class group action.After a brief overview, I will introduce representation varieties and the character ring, discussing the Poisson structure and automorphisms. In particular I will present a new proof of the ergodicity of the mapping class group action on the SU(2)-character variety. Closely related is the action of the modular group on the real Markoff cubic surface $x2 + y2 + z2 - x y z = 2 + t$ and its relationship to Fricke coordinates on Teichmüeller space.
Recently Serge Cantat analyzed the action on the complex Markoff surface, and in particular proved our conjecture on the existence of orbits accumulating both at $SU(2)$-characters and discrete characters.
Other topics which may be discussed in the seminar include:
Teichmueller geodesic flow and Masur's ergodicity theorems for the mapping class groups on spaces of measured foliations
Closely related is the dependence of the (hyper-Kaehler) geometry of the complex moduli space on the Riemann surface's complex structure. In this case, the moduli spaces appear as moduli of holomorphic vector bundles (when the group is compact) and Higgs bundles in general.
Much of the $SU(2)$-theory has been extended to general compact Lie groups K. Pickrell-Xia proved ergodicity for mapping class groups of closed surfaces on K-characters, and Gelander recently established ergodicity for $Out(F_n)$-action on K-character varieties of a free groups $F_n$. Fisher has applied these latter ideas to the spectral gap conjecture.
Closely related work by Iwasaki et al on moduli solutions of Painleve equation, and automorphisms of affine cubic surfaces
Applications to the monodromy question: which surface group representations occur as monodromy representations for projective structures (real, complex, etc.) on closed surfaces? Answering a question posed by Gunning, Gallo-Kapovich-Marden pushed through a program due to Thurston to construct CP1-structures on surfaces with given monodromy. A key point in this proof is the action of the mapping class group on the representation variety. Similar questions exist for other geometric structures related to surfaces.
Properness criteria, energy functional on Teichmueller space for harmonic maps and special surface group representations (Wienhard, Labourie, Wentworth)
For a general (but already outdated!) survey of this subject, see: "Mapping Class Group Dynamics on Surface Group Representations," in "Problems on Mapping Class Groups and Related Topics", B. Farb, ed., Proceedings of Symposia in Pure Math. Proc. Symp. in Pure Math. 74, American Mathematical Society (2006), 189-214. math.GT/0509114