Pure Mathematics Seminar: Caroline Series
Caroline Series, University of Warwick
Wednesday 06 November 2013, 1500-1600
A54, Postgraduate Statistics Centre Lecture Theatre
Representations of the free group into SL(2,C): discreteness and the Bowditch conjecture
It is an old problem to determine when a representation of the free group F2 on two generators into SL(2,C) is free and discrete. Such a representation is determined up to conjugacy by the traces of any triple (X,Y,XY), where X,Y are a pair of generators.
The traces of all possible triples can be neatly calculated by arranging them as the vertices of a trivalent tree, dual to the Farey tesselation of the hyperbolic plane. In 1997, Bowditch introduced a condition on traces which, using some ingenious manipulations on the tree, gave a purely combinatorial proof of McShane's identity, an important result about traces previously only known for discrete groups.
The precise relationship of Bowditch's condition to discrete groups remains rather mysterious. We present some surprising computer graphics which compare the set of representations which satisfy a generalised version of Bowditch's condition with those which are free and discrete. This is joint work with Ser Peow Tan and Yasushi Yamasita.