arXiv subjects

Guillaume Dreyer

Publications and source records attributed to Guillaume Dreyer.

4 recordsLinked to original sources

arXiv2015

Hitchin characters and geodesic laminations

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the surface. This is a natural extension of Thurston's parametrization of the Teichmueller space of S by shear coordinates associated to a maximal geodesic lamination, corresponding to the case n=2. However, significantly new ideas are needed in this higher dimensional case. The article concludes with a few applications.

math.GT↗

arXiv2013

Thurston's cataclysms for Anosov representations

Given an Anosov representation $ρ\colon π_1(S) \to \PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination $λ$ in a surface $S$, we construct shear deformations along the leaves of the geodesic lamination $λ$ endowed with a certain flag decoration, that is provided by the associated flag curve $\mathcal{F}_ρ\colon \Sinf \to \mathrm{Flag}(\mathbb{R}^n)$ of the Anosov representation $ρ$; these deformations generalize to Labourie's Anosov representations Thurston's cataclysms for hyperbolic structures on surfaces. A cataclysm is parametrized by a transverse $n$--twisted cocycle for the orientation cover $\La$ of $λ$. In addition, we establish various geometric properties for these deformations. Among others, we prove a variation formula for the associated length functions $\ell^i_ρ$ of the Anosov representation $ρ$.

math.GT↗

arXiv2013

Length functions of Hitchin representations

Given a Hitchin representation $ρ\colon π_1(S) \to \PSL_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^ρ\colon \mathcal \CH(S) \to \mathbb{R}$ defined on the space of Hölder geodesic currents $\CH(S)$ such that, for a closed, oriented curve $γ$ in $S$, the $i$--th eigenvalue of the matrix $ρ(γ)\in \PSL_n(\mathbb{R})$ is of the form $\pm \mathrm{exp}\, \ell_i^ρ(γ)$: such functions generalize to higher rank Thurston's length function of Fuchsian re\presentations. Identities, diffe\rentiability properties of these lengths $\ell_i^ρ$, as well as applications to eigenvalue estimates, are also considered.

math.GT↗

arXiv2013

Parametrizing Hitchin components

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination λin S with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of λ; and triangle invariants associated with each component of the complement S-λ. We describe identities and relations satisfied by these invariants, and use the resulting coordinates to parametrize the Hitchin component.

math.GT↗