On order preserving representations

TL;DR

This paper introduces order preserving representations of surface groups into Lie groups with bi-invariant orders, showing their properties and connections to maximal representations, with geometric characterizations for Hermitian tube type groups.

Contribution

It establishes the concept of order preserving representations, relates them to weakly maximal representations, and provides geometric insights for Hermitian tube type Lie groups.

Findings

Order preserving representations are faithful with discrete images.

The set of order preserving representations is closed in the representation variety.

Geometric characterization via causal structures on the Shilov boundary for Hermitian tube type groups.

Abstract

In this article we introduce order preserving representations of fundamental groups of surfaces into Lie groups with bi-invariant orders. By relating order preserving representations to weakly maximal representations, introduced in arXiv:1305.2620, we show that order preserving representations into Lie groups of Hermitian type are faithful with discrete image and that the set of order preserving representations is closed in the representation variety. For Lie groups of Hermitian type whose associated symmetric space is of tube type we give a geometric characterization of these representations in terms of the causal structure on the Shilov boundary.