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 ${\rm Sp}(2n,\mathbb F)$. We show that ultralimits of maximal representations in ${\rm Sp}(2n,\mathbb R)$ admit such a framing, and that all maximal framed representations satisfy a suitable generalisation of the classical Collar Lemma. In particular this establishes a Collar Lemma for all maximal representations into ${\rm Sp}(2n,\mathbb R)$. We then describe a procedure to get from representations in ${\rm Sp}(2n,\mathbb F)$ 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.