Positivity and representations of surface groups

TL;DR

This paper introduces the concept of -positive representations of surface groups, proves their key properties, and shows they form open, closed, and connected components in the representation variety for certain Lie groups.

Contribution

It defines -positive representations, proves they are -Anosov, and establishes their topological and geometric properties within the representation space.

Findings

-positive representations are -Anosov.

They are discrete and faithful.

They form open, closed, and connected components in the representation variety.

Abstract

In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $\Theta$-positivity, a generalization of Lusztig's total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $\Theta$-positive representations of surface groups. We prove that $\Theta$-positive representations are $\Theta$-Anosov. This implies that $\Theta$-positive representations are discrete and faithful and that the set of $\Theta$-positive representations is open in the representation variety. We show that the set of $\Theta$-positive representations is closed within the set of representations that do not virtually factor through a parabolic subgroup. From this we deduce that for any simple Lie group $\mathsf G$ admitting a $\Theta$-positive structure there exist components consisting of $\Theta$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $\Theta$-positive representations.