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

Abstract

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 $\hat{I}“$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $\hat{I}“$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in ${\mathbb{H}}^{p,q}$ is a union of connected components of $\mathrm{Hom}(\hat{I}“,\mathrm{PO}(p,q+1))$. More generally, we show that if $\hat{I}“$ 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 $\hat{I}“$ 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 $\mathrm{Hom}(\hat{I}“,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichmüller spaces.

75 pages, 1 figure. To appear in Geom. Funct. Anal