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2019 The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety
Daniele Alessandrini, Brian Collier
Geom. Topol. 23(3): 1251-1337 (2019). DOI: 10.2140/gt.2019.23.1251

Abstract

We describe the space of maximal components of the character variety of surface group representations into PSp ( 4 , ) and Sp ( 4 , ) .

For every real rank 2 Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp ( 4 , ) and Sp ( 4 , ) , 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 PSp ( 4 , ) and Sp ( 4 , ) 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 PSp ( 4 , ) –representations.

Citation

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Daniele Alessandrini. Brian Collier. "The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety." Geom. Topol. 23 (3) 1251 - 1337, 2019. https://doi.org/10.2140/gt.2019.23.1251

Information

Received: 27 August 2017; Accepted: 21 July 2018; Published: 2019
First available in Project Euclid: 5 June 2019

zbMATH: 07079059
MathSciNet: MR3956893
Digital Object Identifier: 10.2140/gt.2019.23.1251

Subjects:
Primary: 22E40, 53C07
Secondary: 14H60, 20H10

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.23 • No. 3 • 2019
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