Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups release_wmm2ow7n6bhkbkxhnk7je2nvme

Abstract

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 Hom(F_r,G)/G admits a strong deformation retraction to the orbit space Hom(F_r,K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G, such as Sp(4,R) and 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.
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