John Parker

Let S be a closed, orientable surface without boundary of genus at least two. In the 1950s Fenchel and Nielsen showed how to give geometrical coordinates to the Teichmueller space of S. We can reinterpret these coordinates algebraically by viewing Teichmueller space as the space of conjugacy classes of discrete, faithful, totally loxodromic, marked representations of the fundamental group of S to PSL(2,R). These coordinates have been generalised to representations to PSL(2,C), SL(3,R) and SU(2,1) by Kourouniotis, Tan, Goldman and Parker-Platis. In this talk, I will generalise to the case of SL(3,C), and show how these coordinates specialise in each of the cases listed above. The talk will be joint work with my PhD student Rodrigo Davila.