3:30 pm Thursday, September 13, 2012
Geometry Seminar: Higher Laminations and Affine Buildings by Ian Le (Northwestern) in RLM 9.166
Let S be a surface, and let G be a split-real semisimple Lie group, for example SL_n(R). We are interested in studying the space of G-bundles on S with framing at the boundary, sometimes called "higher Teichmuller space," which we will denote X_{G,S}. It turns out that this space has a natural tropicalization. When G=SL_2, the space X_{SL_2,S} is a version of Teichmuller space. A remarkable result of Fock and Goncharov tells us that the tropical points of X_{SL_2,S} is precisely the space of measured laminations. Measured laminations were a tool invented by Thurston in the study of two- and three-dimensional topology. This suggests that the tropical points of X_{G,S} should be considered as some kind of space of laminations. Inspired by work of Parreau and others, I will explain how the tropical points of higher Teichmuller space have a geometric description in terms of affine buildings, and that this gives the correct analogue of classical theory of laminations. An important ingredient will be the study of "positive configurations of flags." Time permitting, I will discuss some applications. Submitted by
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