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

52 pages, various tikzpictures

References in corpus (4)

• Ends of the moduli space of Higgs bundles
• Asymptotics of certain families of Higgs bundles in the Hitchin component
• Degeneration of Hitchin representations along internal sequences
• Collar lemma for Hitchin representations

Cited by in corpus (8)

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• Positively ratioed representations
• Groupes convexes-cocompacts en rang supérieur
• Nearly geodesic immersions and domains of discontinuity
• Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$