## Geometry & Topology

### The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety

#### Abstract

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

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 $PSp ( 4 , ℝ )$ and $Sp ( 4 , ℝ )$, 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 $PSp ( 4 , ℝ )$ and $Sp ( 4 , ℝ )$ 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 $PSp ( 4 , ℝ )$–representations.

#### Article information

Source
Geom. Topol., Volume 23, Number 3 (2019), 1251-1337.

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1559700274

Digital Object Identifier
doi:10.2140/gt.2019.23.1251

Mathematical Reviews number (MathSciNet)
MR3956893

Zentralblatt MATH identifier
07079059

#### Citation

Alessandrini, Daniele; Collier, Brian. The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety. Geom. Topol. 23 (2019), no. 3, 1251--1337. doi:10.2140/gt.2019.23.1251. https://projecteuclid.org/euclid.gt/1559700274

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