Compactification d'espaces de représentations de groupes de type fini.

Anne Parreau (1)

1 IF - Institut Fourier

Let $\Gamma$ be a finitely generated group and $G$ be a noncompact semisimple connected real Lie group with finite center. We consider the space $\mathcal X$ of conjugacy classes of reductive representations of $\Gamma$ into $G$. We define the {\it translation vector} of an element $g$ in $G$, with values in a Weyl chamber, as a refinement of the translation length in the associated symmetric space. We construct a compactification of $\mathcal X$, 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 $\Gamma$ on affine buildings with no global fixed point. An analoguous result holds for any reductive group $G$ over a local field.

Mots clés: semisimple Lie group; reductive group; discrete subgroups; moduli space; teichmüller space; representation variety; compactification; symmetric space; asymptotic cone; affine building; euclidean building; valuated field

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Théorie des groupes [math.GR]

Anne Parreau. Compactification d'espaces de représentations de groupes de type fini.. Mathematische Zeitschrift, 2011, pp.1-36. ⟨10.1007/s00209-011-0921-8⟩. ⟨hal-00461218v2⟩