Simple length rigidity for Hitchin representations

Abstract

Nous montrons qu'une représentation de Hitchin est déterminée par les rayons spectraux des images de courbes simples et non séparantes. Comme application, nous caractérisons les isométries de la function d'intersection pour les composantes de Hitchin en dimension 3, ainsi que pour les composantes auto-duales en toutes dimensions. Un outil important de notre démonstration est un résultat de transversalité sur les quadruplets positifs de drapeaux.-------We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in the proof, we establish a transversality result for positive quadruples of flags.

The energy spectrum of metrics on surfaces

“…Let us point out that very closely related results are obtain by Bridgeman, Canary and Labourie in [BCL20]. Namely, they prove that Hitchin representations are rigid for a different type of simple length spectrum 2 .…”

“…Then the spectral length of ρ(γ) is L ρ (γ) = max i=1,...,n |λ i | and its trace is |Tr(ρ(γ))| = n i=1 |λ i |. In [BCL20] it is proved that a Hitchin representation is determined, up to conjugacy, by its simple (spectral) length spectrum (L ρ (γ)) γ∈S and by its simple trace spectrum (|Tr(ρ(γ))|) γ∈S . In contrast, the simple length spectrum we consider in this paper assigns to each simple closed curve γ the translation length of ρ(γ), which is given by ρ (γ) = n i=1 |λ i (ρ(γ))| 2 .…”

“…Note that in[BCL20] the term 'simple length spectrum' is also used; however, it does not refer to the same quantity we consider here.…”

Pressure metrics for cusped Hitchin components

“…If α, β ∈ Γ are hyperbolic, then we say that they have non-intersecting axes if the axes of α and β do not intersect in H 2 . The following strengthened transversality property follows immediately from Theorem 3.6 in [10]. The next lemma collects some of the ingredients needed in Section 7 to study the degeneracy of the Hilbert pressure metric on H 3 (Γ).…”

Finite step Rigidity of Hitchin representations and special Margulis-Smilga spacetimes

“…Similar results involving all elements were also obtained by Kim [Kim01] and Cooper-Delp [CD10] in the context of Hilbert length spectra of projective manifolds. Moreover, in a recent work [BCL20] Bridgeman-Canary-Labourie show that two Hitchin representations are conjugate to each other if and only if they have the same marked trace spectra over simple non-separating elements. Similar results involving infinite data but only over simple non-separating elements were also obtained by Brideman-Canary [BC17] in the context of representations of surface groups inside PSL 2 (C) and by Duchin-Leininger-Rafi [DLR10] in the context of flat surfaces.…”