[Composantes exotiques des variétés de représentations du groupe fondamental d'une surface dans SO(p,q) et les fibrés de Higgs correspondants]
Pour les groupes de Lie semisimples, les espaces de modules de fibrés de Higgs sur une surface de Riemann sont en correspondance avec les variétés de représentations du groupe fondamental de la surface. Pour de nombreux groupes, les invariants topologiques naturels distinguent les composantes connexes de l'espace des modules. Les représentations de Hitchin dans un groupe réel déployé et des représentations maximales dans un groupe hermitien fournissaient les seuls exemples connus jusqu'ici dans lesquels les invariants primitifs étaient insuffisants. Cette note a pour objet d'annoncer l'existence de nouvelles composantes exotiques pour les espaces de modules pour les groupes , pour .
For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups with . These groups lie outside formerly know classes of groups associated with exotic components.
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@article{CRMATH_2018__356_6_666_0,
author = {Aparicio-Arroyo, Marta and Bradlow, Steven and Collier, Brian and Garc{\'\i}a-Prada, Oscar and Gothen, Peter B. and Oliveira, Andr\'e},
title = {Exotic components of {SO(\protect\emph{p},\protect\emph{q})} surface group representations, and their {Higgs} bundle avatars},
journal = {Comptes Rendus. Math\'ematique},
pages = {666--673},
publisher = {Elsevier},
volume = {356},
number = {6},
year = {2018},
doi = {10.1016/j.crma.2018.04.024},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.04.024/}
}
TY - JOUR AU - Aparicio-Arroyo, Marta AU - Bradlow, Steven AU - Collier, Brian AU - García-Prada, Oscar AU - Gothen, Peter B. AU - Oliveira, André TI - Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars JO - Comptes Rendus. Mathématique PY - 2018 SP - 666 EP - 673 VL - 356 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2018.04.024/ DO - 10.1016/j.crma.2018.04.024 LA - en ID - CRMATH_2018__356_6_666_0 ER -
%0 Journal Article %A Aparicio-Arroyo, Marta %A Bradlow, Steven %A Collier, Brian %A García-Prada, Oscar %A Gothen, Peter B. %A Oliveira, André %T Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars %J Comptes Rendus. Mathématique %D 2018 %P 666-673 %V 356 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2018.04.024/ %R 10.1016/j.crma.2018.04.024 %G en %F CRMATH_2018__356_6_666_0
Aparicio-Arroyo, Marta; Bradlow, Steven; Collier, Brian; García-Prada, Oscar; Gothen, Peter B.; Oliveira, André. Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars. Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 666-673. doi : 10.1016/j.crma.2018.04.024. http://www.numdam.org/articles/10.1016/j.crma.2018.04.024/
[1] The Geometry of -Higgs Bundles, Departamento de Matemáticas, Facultad de Ciencias de la Universidad de Salamanca, 2009 (Ph.D. Thesis)
[2] -Higgs bundles and higher Teichmüller components (preprint) | arXiv
[3] Cayley and Langlands type correspondences for orthogonal Higgs bundles (preprint) | arXiv
[4] Higgs bundles, the Toledo invariant and the Cayley correspondence, J. Topol., Volume 10 (2017), pp. 795-826
[5] Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedic., Volume 122 (2006), pp. 185-213
[6] Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q., Volume 1 (2005), pp. 555-601 (No. 3, Special Issue: In memory of Armand Borel. Part 2)
[7] Higher Teichmüller spaces: from SL(2, ) to other Lie groups (Papadopoulos, A., ed.), Handbook of Teichmüller Theory, vol. IV, 2012, pp. 539-618
[8] -surface group representations and Higgs bundles (preprint) | arXiv
[9] The geometry of maximal representations of surface groups into (preprint) | arXiv
[10] Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci., Volume 103 (2006), pp. 1-211
[11] The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations (preprint) | arXiv
[12] Components of spaces of representations and stable triples, Topology, Volume 40 (2001), pp. 823-850
[13] Positivity and higher Teichmüller theory, Proceedings of the 7th European Congress of Mathematics, 2016
[14] Lie groups and Teichmüller space, Topology, Volume 31 (1992), pp. 449-473
[15] Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006), pp. 51-114
[16] Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2008
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