Filippo Mazzoli, University of Virginia
Abstract: The focus of this talk will be the study of representations of the fundamental group of a closed orientable surface of genus g (in short, a surface group) inside the Lie groups PSL(n,C), for any g and n larger than 1. Surface group representations inside PSL(2,C) (such as quasi-Fuchsian representations) arise from and are of interest for many areas in mathematics, such as the study of Riemann surfaces, 2 and 3-dimensional hyperbolic geometry, and complex dynamics. Since the works of Labourie, Guichard-Wienhard, and Kapovich-Leeb-Porti, many of the techniques deployed in these contexts are now finding novel applications to study wide classes of surface group representations inside higher-rank Lie groups, called “Anosov representations”.
After discussing the main concepts and motivations, I will talk about an upcoming joint work with S. Maloni, G. Martone, and T. Zhang, where we develop holomorphic coordinates of (suitable open subsets of) of the space of surface group representations inside PSL(n,C), by generalizing the works of Bonahon and Thurston on pleated surfaces in hyperbolic 3-manifolds.