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

Abstract

For any integers $p\geq 2$ and $q\geq 1$, let $\mathbb{H}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $\Gamma$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $\Gamma$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(\Gamma,\mathrm{PO}(p,q+1))$. More generally, we show that if $\Gamma$ 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 $\Gamma$ to $\mathrm{PO}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(\Gamma,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichm\"uller spaces.