Maximal representations, non-archimedean siegel spaces, and buildings

Let 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(2n; F). We show that ultralimits of maximal representations in Sp(2n; R) 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(2n; R). We then describe a procedure to get from representations in Sp(2n; 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.