Seminar

When S is a compact surface and G < PSL(2,R) is a Fuchsian group such that  S = H^2/G , it is a well-known classical fact that the connected component of the set of discrete and faithful representations G -> PSL(2,R)  which contains G is actually a full connected component of the representation variety. As such, it is cut out by polynomial inequalities over R. This is far from true when PSL(2,R) is replaced by PSL(2,C) : the quasi-Fuchsian locus of S is embedded in a topologically wild way into Hom(G,PSL(2,C))//PSL(2,C). We will describe some semi-algebraic approximations, and some countable semi-algebraic decompositions, of the set of discrete and faithful PSL(2,C)-representations of geometrically finite Kleinian groups. Some of this work is joint with Gaven J. Martin and Jeroen Schillewart.