No 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 X (Γ , G) of conjugacy classes of semisimple representations of Γ into G, which is the maximal Hausdorff quotient of Hom(Γ , G)/G. We define the translation vector of g ∈ 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 X (Γ , 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.

The real spectrum compactification of character varieties: characterizations and applications