Exploring a family of Kleinian groups : Mumford's discreteness problem (Women in Mathematics)

Abstract

A Kleinian group is a discrete group of linear fractional transformations (Möbius maps) acting on the Riemann sphere. Given two Möbius maps, how can we tell if the group they generate is discrete? Answering this question leads through Riemann surfaces and Teichmüller theory to some of Thurston's wonderful ideas about hyperbolic 3- manifolds. Also involved are some beautiful computer graphics. I will explain these connections through an example originally introduced by David Mumford and discussed in our book Indra's Pearls (D. Mumford, C. Series and D. Wright, Cambridge University Press 2002).