Homotopy type of free group character varieties
  Let G be a real reductive algebraic group with maximal compact subgroup K, and let F be a rank r free group. Here, we summarize the construction of a natural strong deformation retraction from the space of closed orbits in Hom(F,G)/G to the orbit space Hom(F,K)/K. In particular, these spaces have the same homotopy type. 
Ana Casimiro, Carlos Florentino, Sean Lawton and Andr\'e Oliveira
math.AT math.AG math.RT