Positivity and representations of surface groups

• Verfasst von: Guichard, Olivier [VerfasserIn]
• Verfasst von: Labourie, François [VerfasserIn]
• Verfasst von: Wienhard, Anna [VerfasserIn]
• Titel: Positivity and representations of surface groups
• Verf.angabe: Olivier Guichard, François Labourie, and Anna Wienhard
• E-Jahr: 2021
• Jahr: 16 Aug 2021
• Umfang: 42 S.
• Fussnoten: Gesehen am 23.09.2022
• Titel Quelle: Enthalten in: De.arxiv.org
• Ort Quelle: [S.l.] : Arxiv.org, 1991
• Jahr Quelle: 2021
• Band/Heft Quelle: (2021), Artikel-ID 2106.14584, Seite 1-42
• DOI: doi:10.48550/arXiv.2106.14584
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 11-XX, 14-XX
• Sach-SW: Mathematics - Differential Geometry
• Sach-SW: Mathematics - Geometric Topology
• Sach-SW: Mathematics - Group Theory
• K10plus-PPN: 1817280511

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.