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The
geometry of maximal components of the $\mathsf{PSp}(4,\mathbb
R)$ character variety
Daniele Alessandrini and Brian Collier |
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Geometry & Topology 23 (2019) 1251–1337
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Abstract |
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We describe the space of maximal components of the character variety of surface group representations into and . For every real rank Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups and , we give a mapping class group invariant parametrization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular: we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of and by the action of the mapping class group as a holomorphic submersion over the moduli space of curves. These results are proven in two steps: first we use Higgs bundles to give a nonmapping class group equivariant parametrization, then we prove an analog of Labourie’s conjecture for maximal –representations. |
Keywords
character varieties, mapping class group, Higgs bundles,
maximal representations
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Mathematical Subject Classification 2010
Primary: 22E40, 53C07
Secondary: 14H60, 20H10
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References |
Publication
Received: 27 August 2017
Accepted: 21 July 2018
Published: 28 May 2019
Proposed: Benson Farb
Seconded: Jean-Pierre Otal, Anna Wienhard
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Authors |
| Daniele Alessandrini | |
| Mathematisches Institut Universitaet Heidelberg Heidelberg Germany |
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| Brian Collier | |
| Department of Mathematics University of Maryland College Park, MD United States |
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