Prof. Caroline Series (University of Warwick)
In the late 1970s, Robert Riley made extensive computer explorations of a family of subgroups of SL(2,C) generated by the two linear fractional transformations z—> z+1 and z —> z/(cz+1), where c is a complex parameter. In doing so he produced some amazing computer graphics to illustrate the behaviour of these groups for varying c. Among his discoveries was that when c is the cube root of -1, the associated hyperbolic 3-manifold is the complement of the figure of eight knot. This was one of the examples which inspired Thurston’s remarkable hyperbolization theorem.
In this talk we revisit Riley’s computer graphics from the view point of the so-called Keen-Series rays. After explaining the basics and the ray picture, we will go on to show how recent work is enhancing and confirming Riley’s picture.
Meeting ID: 980 7475 6230