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Convex foliated projective structures and the Hitchin component for ${\rm PSL}_4(\mathbf{R})$
Olivier Guichard and Anna Wienhard
Source: Duke Math. J. Volume 144, Number 3 (2008), 381-445.
Abstract
In this article, we give a geometric interpretation of the Hitchin component $\mathcal{T}^4(\Sigma) \subset {\rm Rep}(\pi_1(\Sigma), {\rm PSL}_4(\mathbf{R}))$ of a closed oriented surface of genus $g\geq 2$. We show that representations in $\mathcal{T}^4(\Sigma)$ are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of $\Sigma$. From this, we also deduce a geometric description of the Hitchin component $\mathcal{T}(\Sigma, {\rm Sp}_4(\mathbf{R}))$ of representations into the symplectic group
Primary Subjects: 57M50
Secondary Subjects: 20H10
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1218811400
Digital Object Identifier: doi:10.1215/00127094-2008-040
Mathematical Reviews number (MathSciNet):
MR2444302
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