Parabolic-preserving deformations of cusped hyperbolic lattices

TL;DR

This paper investigates deformations of cusped hyperbolic lattices into complex hyperbolic and higher-dimensional groups, identifying conditions for discreteness and faithfulness, and providing explicit examples of such deformations.

Contribution

It introduces the concept of strongly parabolic-preserving deformations and demonstrates their existence for specific groups like the figure-eight knot group and Bianchi groups.

Findings

The figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into SU(3,1).

Infinitely many bending deformations of Bianchi groups are strongly parabolic-preserving in SU(3,1).

Existence of infinitely many non-commensurable hyperbolic n-manifolds with parabolic-preserving deformations into SU(n,1).

Abstract

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) 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 ${\rm SU}(3,1)$, with further deformations into ${\rm SU}(2,2)$. We also study the \emph{bending deformations} of the Bianchi groups (seen as subgroups of ${\rm SO}(3,1)$) along the modular surface into ${\rm SU}(3,1)$ and ${\rm SO}(4,1)$, and show that infinitely many of them are strongly parabolic-preserving in ${\rm SU}(3,1)$, while none are strongly parabolic-preserving in ${\rm SO}(4,1)$. Finally, for any $n \geqslant 3$, 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 ${\rm SU}(n,1)$.