Compactification d'espaces de représentations de groupes de type fini

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

2010-03-04

arxiv.org/abs/1003.1111v2

Mathematische Zeitschrift, 14 September (2011), pages 1-36

Mathematics

Differential Geometry

40 pages

Scientific paper

10.1007/s00209-011-0921-8

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 Teichm\"uller 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.

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