On weakly maximal representations of surface groups

TL;DR

This paper introduces weakly maximal representations of surface groups into Hermitian Lie groups, proving their discreteness, injectivity, and analyzing their algebraic and geometric properties within the representation variety.

Contribution

It defines weakly maximal representations, establishes their fundamental properties, and explores their structure and relation to other significant subsets in the representation variety.

Findings

Weakly maximal representations are discrete and injective.

The set of weakly maximal representations is closed in the representation variety.

The structure of the Zariski closure of their images is characterized.

Abstract

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representation variety.