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.

arXiv comment52 pages, various tikzpictures

References in corpus (8)

• Ends of the moduli space of Higgs bundles
• Higher Laminations and Affine Buildings
• Asymptotics of certain families of Higgs bundles in the Hitchin component
• Compactification d'espaces de représentations de groupes de type fini
• The degeneration of convex RP^2 structures on surfaces
• Degeneration of Hitchin representations along internal sequences
• Collar lemma for Hitchin representations
• Tropicalization of group representations

Cited by in corpus (8)

• The geometry of maximal representations of surface groups into SO(2,n)
• On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
• Currents, Systoles, and Compactifications of Character Varieties
• Schottky groups and maximal representations
• Positively ratioed representations
• Groupes convexes-cocompacts en rang supérieur
• Nearly geodesic immersions and domains of discontinuity
• Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$