Boundary of Anosov coaffine representations

Abstract

Let $Γ\le \mathrm{SL}(d+1,\mathbb R)$ be a torsion-free hyperbolic group dividing a properly and strictly convex domain $Ω\subset \mathbb R\mathrm P^d$, and let $ρ: Γ\to \mathrm{SL}(d+1,\mathbb R)$ be the inclusion. We study representations of $Γ$ into the coaffine group that are deformations of $ρ\oplus 1 \in \mathrm{Hom}(Γ, \mathrm{GL}(d+2,\mathbb R))$ and that lie on the boundary of the locus of projective Anosov representations. In this setting, we identify a canonical limit set $Λ$ for the action and study its fine geometric features which are, in a suitable sense, stable. We prove the existence of an ending lamination that encodes the failure of the Anosov condition and the non-existence of a Cannon--Thurston map. Under certain conditions, we compute the Hausdorff dimension and show that the corresponding measure is positive.