Positivity and representations of surface groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866915788656476160 |
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| author | Guichard, Olivier Labourie, François Wienhard, Anna |
| author_facet | Guichard, Olivier Labourie, François Wienhard, Anna |
| contents | In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Θ$-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 $Θ$-positive representations of surface groups. We prove that $Θ$-positive representations are $Θ$-Anosov. This implies that $Θ$-positive representations are discrete and faithful and that the set of $Θ$-positive representations is open in the representation variety. We show that the set of $Θ$-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 $Θ$-positive structure there exist components consisting of $Θ$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Θ$-positive representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_14584 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Positivity and representations of surface groups Guichard, Olivier Labourie, François Wienhard, Anna Differential Geometry Group Theory Geometric Topology 11-XX, 14-XX In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Θ$-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 $Θ$-positive representations of surface groups. We prove that $Θ$-positive representations are $Θ$-Anosov. This implies that $Θ$-positive representations are discrete and faithful and that the set of $Θ$-positive representations is open in the representation variety. We show that the set of $Θ$-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 $Θ$-positive structure there exist components consisting of $Θ$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Θ$-positive representations. |
| title | Positivity and representations of surface groups |
| topic | Differential Geometry Group Theory Geometric Topology 11-XX, 14-XX |
| url | https://arxiv.org/abs/2106.14584 |