Invariants of ${\rm PSL}_n\mathbb{R}$-Fuchsian Representations and a Slice of Hitchin Components

J. Math. Sci. Univ. Tokyo
Vol. 28 (2021), No. 4, Page 593-639.

Inagaki, Yusuke
Invariants of ${\rm
PSL}_n\mathbb{R}$-Fuchsian Representations and a Slice of Hitchin Components

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Abstract:
The Hitchin component $H_n(S)$ is a special component of the ${\rm PSL}_n\mathbb{R}$-character variety of a closed surface $S$ of genus $g \geq 2$ which contains the discrete faithful representations $\pi_1(S) \rightarrow {\rm PSL}_2\mathbb{R}$ via an irreducible representation. Bonahon-Dreyer ([BD14], [BD17]) gave a parameterization of $H_n(S)$ by the triangle invariants and the shearing-type invariants fixing an arbitrary maximal geodesic lamination on $S$, so that the Hitchin component is a cone in a Euclidean space.
The images of discrete faithful representations $\pi_1(S) \rightarrow {\rm PSL}_2\mathbb{R}$ in $H_n(S)$ are called ${\rm PSL}_n\mathbb{R}$-Fuchsian representations. In this paper we characterize the ${\rm PSL}_n\mathbb{R}$-Fuchsian representations of the Hitchin component in the Bonahon-Dreyer coordinates. In particular this explicit characterization implies the set of the ${\rm PSL}_n\mathbb{R}$-Fuchsian representations is an affine slice. We also discuss the case when $S$ has boundary.

Keywords: Hitchin components, Teichmüller spaces.

Mathematics Subject Classification (2010): Primary 57N16; Secondary 22E40.
Received: 2020-07-28