Volume 21, issue 6 (2017)

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Maximal representations, non-Archimedean Siegel spaces, and buildings

Marc Burger and Maria Beatrice Pozzetti

Geometry & Topology 21 (2017) 3539–3599
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 $Sp\left(2n,\mathbb{F}\right)$. We show that ultralimits of maximal representations in $Sp\left(2n,ℝ\right)$ 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 $Sp\left(2n,ℝ\right)$. We then describe a procedure to get from representations in $Sp\left(2n,\mathbb{F}\right)$ 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.

Keywords
maximal representation, affine building, collar lemma, real closed field, non-Archimedean symmetric spaces
Mathematical Subject Classification 2010
Primary: 20-XX, 22E40
Publication
Received: 4 November 2015
Revised: 15 October 2016
Accepted: 19 January 2017
Published: 31 August 2017
Proposed: Ian Agol
Seconded: Leonid Polterovich, Yasha Eliashberg
Authors
 Marc Burger Department of Mathematics ETH Zentrum Zürich Switzerland Maria Beatrice Pozzetti Mathematics Institute University of Warwick Coventry United Kingdom