Positivity and representations of surface groups

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.

There was a wrong statement in the appendix of the previous versions. The statement of the main result has therefore to be altered

References in corpus (8)

• Anosov representations: Domains of discontinuity and applications
• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Anosov representations and proper actions
• Higgs bundles and surface group representations in the real symplectic group
• Higgs bundles, the Toledo invariant and the Cayley correspondence
• SO(p,q)-Higgs bundles and higher Teichmüller components
• Positive surface group representations in PO(p,q)
• A general Cayley correspondence and higher Teichmüller spaces

Cited by in corpus (1)

• Fiber bundles associated with Anosov representations