Let $G$ be a real reductive algebraic group with maximal compact subgroup $K$, and let $F_r$ be a rank $r$ free group. We show that the space of closed orbits in $\mathrm{ Hom}(F_r,G)/G$ admits a strong deformation retraction to the orbit space $\mathrm{ Hom}(F_r,K)/K$. In particular, all such spaces have the same homotopy type. We compute the Poincar\'e polynomials of these spaces for some low rank groups $G$, such as $\mathrm{ Sp}(4,\mathbb{ R})$ and $\mathrm{ U}(2,2)$. We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.
Categories: math.AT math.AG math.RT