Weakly maximal representations of surface groups

TL;DR

This paper introduces weakly maximal representations of surface groups into Hermitian Lie groups, extending maximal representations, and explores their properties, including discreteness, injectivity, and topological characterizations.

Contribution

It defines a new class of representations called weakly maximal, extending the concept of maximal representations, and analyzes their algebraic and topological properties.

Findings

Weakly maximal representations are discrete and injective.

They can be characterized by bi-invariant orderings.

In Hermitian tube type groups, orderings relate to causal structures on the Shilov boundary.

Abstract

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are discrete and injective and describe the structure of the Zariski closure of the image. An interesting feature of these representations is that they admit an elementary topological characterization in terms of bi-invariant orderings. In particular if the target group is Hermitian of tube type, the ordering can be described in terms of the causal structure on the Shilov boundary.