Mathematische Zeitschrift

, Volume 272, Issue 1–2, pp 51–86 | Cite as

Compactification d’espaces de représentations de groupes de type fini

  • Anne Parreau


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.


Moduli spaces of representations Higher teichmüller theory Reductive groups Symmetric spaces Euclidean buildings Asymptotic cones 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institut Fourier, Université Grenoble I et CNRSSaint-Martin-d’Hères CedexFrance

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