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Positivity and Representations of Surface Groups by Guichard, Olivier; Labourie, François; Wienhard, Anna is a scholarly article available to read on EtoBox.

What is Positivity and Representations of Surface Groups about?

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.

Author

Guichard, Olivier; Labourie, François; Wienhard, Anna

Published

2021

Language

EN