Boundary of Anosov coaffine representations

Abstract

Let $\Gamma \le \mathrm{SL}(d+1,\mathbb{R})$ be a torsion-free hyperbolic group dividing a properly and strictly convex domain $\Omega \subset \mathbb{R}{\mathrm{P}}^d$, and let $\rho :\Gamma \to \mathrm{SL}(d+1,\mathbb{R})$ be the inclusion. We study representations of $\Gamma$ into the coaffine group that are deformations of $\rho \oplus 1\in \mathrm{Hom}(\Gamma ,\mathrm{GL}(d+2,\mathbb{R}))$ and that lie on the boundary of the locus of projective Anosov representations. In this setting, we identify a canonical limit set $\Lambda$ 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.

arXiv comment80 pages, 8 figures