Maximal representations, non-Archimedean Siegel spaces, and buildings

Abstract

Abstract. 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.

Collar lemma for Hitchin representations

“…This analog of the Margulis lemma was later extended by Cooper-Long-Tillman [9] to all convex real projective manifolds. Burger-Pozzetti [5] also recently proved a similar result for maximal representations into Sp(2n, R).…”

Positivity and higher Teichmüller theory

“…We refer the reader to [12,50,29,13] for more details and further properties of maximal representations.…”

Geometric Structures and Representations of Discrete Groups

“…(1) the proper discontinuity of the action of the mapping class group Mod(S) [86]; (5) versions of the collar lemma for associated locally symmetric spaces [29,89]. Other higher Teichmüller spaces of Anosov representations of π 1 (S) are also being explored [70].…”