Topology and singularities of free group character varieties
We will discuss some generalities of the geometry, topology and singularities of the the G-character variety of F, that is, the moduli space Hom(F,G)/G of representations of a finitely presented group F into a Lie group G.
Then, we concentrate on the case when G is a complex affine reductive Lie group with maximal compact subgroup K, and F is a free group of rank r. In this situation, it can be proved that Hom(F,K)/K is a strong deformation retract of Hom(F,G)/G; in particular, both spaces have the same homotopy type. In the case G=SL(n,C), one can explicitly describe the singular locus of these character varieties, showing that they have the homotopy type of a manifold only when F or G are abelian, or r+n<6. In the non-abelian case and r+n=5, the varieties have the homotopy type of spheres. This is joint work with Sean Lawton.