Parabolic-preserving deformations of cusped hyperbolic lattices

Abstract

We study deformations of non-cocompact lattices of $\mathrm{SO}(n,1)$ into $\mathrm{SU}(n,1)$ and $\mathrm{SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n⩾3$) is for the parabolic subgroups to remain parabolic and discrete; we call such representations \emph{strongly parabolic-preserving}. We show that the figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into $\mathrm{SU}(3,1)$, with further deformations into $\mathrm{SU}(2,2)$. We also study the \emph{bending deformations} of the Bianchi groups (seen as subgroups of $\mathrm{SO}(3,1)$) along the modular surface into $\mathrm{SU}(3,1)$ and $\mathrm{SO}(4,1)$, and show that infinitely many of them are strongly parabolic-preserving in $\mathrm{SU}(3,1)$, while none are strongly parabolic-preserving in $\mathrm{SO}(4,1)$. Finally, for any $n⩾3$, we show that there exist infinitely many non-commensurable cusped hyperbolic $n$-manifolds whose corresponding hyperbolic representation admits a 1-parameter family of parabolic-preserving deformations into $\mathrm{SU}(n,1)$.

Comments welcome! This paper supercedes arXiv:1702.00508