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arXiv · 1403.3603

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

Ana Casimiro·Carlos Florentino·Sean Lawton·André Oliveira
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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.

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Ana Casimiro, Carlos Florentino, Sean Lawton, André Oliveira. 2014-07-02. Topology of Moduli Spaces of Free Group Representations in Real Reductive Groups. https://doi.org/10.1515/forum-2014-0049

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