The Topology of Parabolic Character Varieties of Free Groups

Abstract

Let G be a complex affine algebraic reductive group, and let K be a maximal compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or equal to 0, let X (respectively, Y) be the space of equivalence classes of representations of the free group of m+n generators in G (respectively, K) such that for each i between 1 and m, the image of the i-th free generator is conjugate to h_i. These spaces are parabolic analogues of character varieties of free groups. We prove that Y is a strong deformation retraction of X. In particular, X and Y are homotopy equivalent. We also describe explicit examples relating X to relative character varieties.