Compactification d'espaces de repr\'esentations de groupes de type fini

TL;DR

This paper introduces a new compactification of the space of conjugacy classes of reductive group representations of finitely generated groups into noncompact semisimple Lie groups, extending Thurston's work to higher rank groups and affine buildings.

Contribution

It constructs a compactification of the representation space using marked translation vector spectra, generalizing Thurston's compactification to higher rank Lie groups and affine buildings.

Findings

Boundary points correspond to actions on affine buildings without fixed points

The compactification generalizes Thurston's for Teichmüller space

Applicable to reductive groups over local fields

Abstract

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.