Maximal representations, non-Archimedean Siegel spaces, and buildings

Abstract

Let $\mathbb{F}$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in $Sp\left(2n,\mathbb{F}\right)$. We show that ultralimits of maximal representations in $Sp\left(2n,ℝ\right)$ admit such a framing, and that all maximal framed representations satisfy a suitable generalization of the classical collar lemma. In particular, this establishes a collar lemma for all maximal representations into $Sp\left(2n,ℝ\right)$. We then describe a procedure to get from representations in $Sp\left(2n,\mathbb{F}\right)$ interesting actions on affine buildings, and in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.