【Seminar】Fenchel-Nielsen coordinates for SL(3,C) representations of closed surface groups

Speaker: Prof. John Parker(Durham University)

Title: Fenchel-Nielsen coordinates for SL(3,C) representations of closed surface groups

Abstract:

In the 1940’s Fenchel and Nielsen gave geometrical coordinates on the Teichmueller space of a closed surface of genus at least two, or equivalently on discrete, faithful, totally loxodromic representations of its fundamental group to PSL(2,R). These coordinates have been generalised to quasi-Fuchsian representations in PSL(2,C), by Kourouniotis and Tan; to convex real projective representations to SL(3,R), by Goldman and later Zhang; and to complex hyperbolic representations to SU(2,1) by Platis and me. In this talk we describe how to give Fenchel-Nielsen coordinates to SL(3,C) representations of surface groups to SL(3,C) and we show how this includes all the previous constructions as special cases, thereby unifying them. This is work from the PhD thesis of my student Rodrigo Davila.