$C^1$-rigidity of $\mathbb{H}^{p,q}$-convex cocompact representations of Gromov hyperbolic groups

Abstract

Let $p>3$ and let $Γ$ be a non-elementary Gromov hyperbolic group with $\partial_\inftyΓ\cong\mathbb S^{p-1}$. We prove that every $\mathbb{H}^{p,2}$-convex cocompact representation $$ ρ:Γ\longrightarrow \mathrm{PO}_{p,3} $$ whose proximal limit set $Λ_ρ\subset\partial_\infty\mathbb{H}^{p,2}$ is a $C^1$ submanifold preserves a totally geodesic copy of $\mathbb{H}^p\subset\mathbb{H}^{p,2}$. Consequently, $Γ/\ker ρ$ is virtually a cocompact lattice in $\mathrm{Isom}(\mathbb{H}^p)$. For odd $p$, we prove the same conclusion for every signature $(p,q)$ with $p>q\ge2$, under the same hypothesis $\partial_\inftyΓ\cong\mathbb S^{p-1}$. Our methods combine dynamics, differential topology, and representation theory. The key new ingredient is to study the radicals of the semidefinite forms induced on the tangent spaces of the limit set. They form a continuous vector bundle of constant rank. The tangent action at attracting fixed points gives a bound, in terms of the radical rank, on the number of eigenvalues of an orthogonal lift whose modulus exceeds one. We then use the topology of the sphere and the representation theory of real algebraic groups to exclude the remaining possibilities. In arbitrary signature with $p>q$, these arguments already imply that a representation with a $C^1$ limit set cannot be Zariski dense.