Maximal representations, non Archimedean Siegel spaces, and buildings

TL;DR

This paper introduces the concept of maximal framing for representations into symplectic groups over real closed fields, extends classical geometric lemmas, and explores actions on affine buildings, revealing structural properties of certain group elements.

Contribution

It defines maximal framing for representations over real closed fields, generalizes the Collar Lemma, and links maximal representations to actions on affine buildings, providing new insights into their structure.

Findings

Ultralimits of maximal representations admit maximal framing.

Maximal framed representations satisfy a generalized Collar Lemma.

Elements with zero translation length have a specific structure in these representations.

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