Geometric structures and surface group representations into Lie groups

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The proposed projects aim to develop geometry of representations of fundamental groups of surfaces S (surface groups) into Lie groups. Finitely generate discrete groups of PSL(2, R) and PSL(2, C), called Kleinian groups, have fruitfully studied, especially, last a couple of decades. They are the groups of orientation preserving isometries of the hyperbolic spaces of dimension two and three, respectively, and represent negatively curved geometry. They play central roles in the Teichmüller theory, the study of Riemann surfaces, and 3-dimensional hyperbolic manifolds, in particular, in relation with geometrization of 3-dimensional topological manifolds. The theme of my proposed projects is to generalize interesting properties of Kleinian groups and Teichmüller theory, to more general representations and to relate geometry, topology, algebra, analysis and dynamics. One research direction of this proposal is to understand geometry of representations of the surface group into PSL(2, R) or PSL(2, C) whose images are non-discrete subgroups (non-discrete representations). Those projects seek for the geometric meaning of non-discrete representations. In particular, they are closely related to the dynamics of the action of map- ping class groups on the character varieties, the space of surface group representations. Another research direction is to study representations of surface groups to higher rank Lie groups, in particular, PSL(3, R) and PSL(3, C). While rank-one Lie groups correspond to "negatively curved geometry", higher rank Lie groups correspond to "non-positively curved geometry" and give rich geometry. It is an increasingly active area of research, and my projects concern basic problems of higher-rank representations.