A collar lemma for partially hyperconvex surface group representations

Abstract

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into S L ( n , R ) SL(n,\mathbb {R}) that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for Θ \Theta -positive representations into S O ( p , q ) SO(p,q) if p ≥ 4 p\geq 4 . We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.

Simple length rigidity for Hitchin representations

“…Following Martone-Zhang [19], say a projective Anosov representation ρ admits a positive cross ratio if 0 < Tr(p ρ (g)p ρ (h)) < 1 for any two intersecting geodesics g and h. Examples come from Teichmüller spaces and Hitchin representations [14,19]. More generally positive representations are associated to positive cross ratios [2]. Our first application is a generalisation of the convexity theorem of Kerckhoff [11] and was the initial reason for our investigation:…”

“…Let µ be a (ρ, w)-integrable Θ-current. Then for every y in H2 , Ω ρ(G) belongs to L 1 (G p , µ). Moreover…”

Nearly geodesic immersions and domains of discontinuity

“…Representations satisfying the Θ-positive property were studied by Beyer and Pozzetti [BP21]. In particular they show that all Θ-positive representations ρ : Γ g Ñ PSOpp, qq can be deformed to a h Θ -generalized Fuchsian representation when q ą p `1, so in this case Corollary 8.11 applies to all Θ-positive representations.…”

“…More generally, Guichard and Wienhard defined a notion of Θ-positive representations for some simple Lie groups and some subsets of simple roots Θ, of which Hitchin and maximal representations are a particular instance. Beyer and Pozzetti proved that the spaces of Θ-positive representations in SOpp, qq for 3 ď p ă q are higher rank Teichmüller spaces [BP21]. Bradlow, Collier, Garcia-Prada, Gothen, and Oliveira [BCGP `21] and Guichard, Labourie and Wienhard [GLW21] proved that for each notion of Θ-positivity there exist higher rank Teichmüller spaces consisting of Θpositive representations.…”

Positive crossratios, barycenters, trees and applications to maximal representations

“…To deduce Theorems A and B, from Theorem D and Corollary E, we use the maximal framing to construct a positive crossratio [ • , • , • , • ] ρ on H Γ whose periods satisfy the equality per(γ) = L(ρ(γ)). Maximal representations are not the only class of representations whose length function is given by the periods of a positive crossratio: this is the case for all positively ratioed representations [MZ19], a class that also includes Hitchin representations [Lab07], representations satisfying property H k [BP20] and Θ-positive representations [BP21]. Corollary E can be used to study asymptotic properties of these representations as well.…”