Maximal representations, non Archimedean Siegel spaces, and buildings

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 $\mathrm{S}\mathrm{p}(2n,\mathbb{F})$. We show that ultralimits of maximal representations in $\mathrm{S}\mathrm{p}(2n,\mathbb{R})$ 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 $\mathrm{S}\mathrm{p}(2n,\mathbb{R})$. We then describe a procedure to get from representations in $\mathrm{S}\mathrm{p}(2n,\mathbb{F})$ 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.