Maximal representations, non-Archimedean Siegel spaces, and buildings

Burger, Marc; Pozzetti, Maria Beatrice

doi:10.2140/gt.2017.21.3539

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Maximal representations, non-Archimedean Siegel spaces, and buildings

Marc Burger and Maria Beatrice Pozzetti

Geometry & Topology 21 (2017) 3539–3599

DOI: 10.2140/gt.2017.21.3539

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

Keywords

maximal representation, affine building, collar lemma, real closed field, non-Archimedean symmetric spaces

Mathematical Subject Classification 2010

Primary: 20-XX, 22E40

References

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

Volume 21, issue 6 (2017)