${\mathbb{H}}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces

For any integers $p\ge 2$ and $q\ge 1$, let ${\mathbb{H}}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $\mathrm{\Gamma }$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $\mathrm{\Gamma }$ to $\mathrm{P}\mathrm{O}(p,q+1)$ which are convex cocompact in ${\mathbb{H}}^{p,q}$ is a union of connected components of $\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },\mathrm{P}\mathrm{O}(p,q+1))$. More generally, we show that if $\mathrm{\Gamma }$ is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension $p$, then the set of injective and discrete representations of $\mathrm{\Gamma }$ to $\mathrm{P}\mathrm{O}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of ${\mathbb{H}}^{p,q}$ is a union of connected components of $\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{\Gamma },\mathrm{P}\mathrm{O}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichmüller spaces.