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Compactification d’espaces de représentations de groupes de type fini

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Abstract

Let Γ be a finitely generated group and G be a noncompact semisimple connected real Lie group with finite center. We consider the space \({\fancyscript X(\Gamma, G)}\) of conjugacy classes of semisimple representations of Γ into G, which is the maximal Hausdorff quotient of \({{\rm Hom}(\Gamma, G)/G}\) . We define the translation vector of \({g \in G}\), with value in a Weyl chamber, as a natural refinement of the translation length of g in the symmetric space associated with G. We construct a compactification of \({\fancyscript X(\Gamma, G)}\) , induced by the marked translation vector spectrum, generalizing Thurston’s compactification of the Teichmüller space. We show that the boundary points are projectivized marked translation vector spectra of actions of Γ on affine buildings with no global fixed point. An analoguous result holds for any reductive group G over a local field.

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  1. Institut Fourier, Université Grenoble I et CNRS, BP 74, 38402, Saint-Martin-d’Hères Cedex, France

    Anne Parreau

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Correspondence to Anne Parreau.

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avec le soutien de l’ANR Repsurf: ANR-06-BLAN-0311.

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Parreau, A. Compactification d’espaces de représentations de groupes de type fini. Math. Z. 272, 51–86 (2012). https://doi.org/10.1007/s00209-011-0921-8

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