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

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 $\mathrm{Hom}(F_r,G)/G$ 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 Poincaré 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.

v2: exposition improved, typos corrected, and a minor gap in a proof fixed; 25 pages; accepted at Forum Mathematicum

References in corpus (2)

• The topology of moduli spaces of free group representations
• The topology of nilpotent representations in reductive groups and their maximal compact subgroups

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• Dimension of representation and character varieties for two and three-orbifolds
• Flawed groups and the topology of character varieties
• Poisson maps between character varieties: gluing and capping
• Geometry of $\mathrm{SU}(3)$-character varieties of torus knots
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• Dynamics on the SU(2,1)-character variety of the one-holed torus
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