Geometry and Topology Seminar: Geometric Structures for the G_2’ Hitchin Component

Abstract: Fundamental to our understanding of Teichmüller space T(S) of a closed oriented genus g ≥ 2 surface S are two different perspectives: one as connected component in the PSL(2,\R) character variety \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2. After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves that serve as bridge between the geometric structures and representations.

Host: Charles Ouyang