Rank 1 deformations of non-cocompact hyperbolic lattices

Abstract

Let $X$ be a negatively curved symmetric space and $\Gamma$ a non-cocompact lattice in $\mathrm{Isom}(\mathrm{X})$. We show that small, parabolic-preserving deformations of $\Gamma$ into the isometry group of any negatively curved symmetric space containing $X$ remain discrete and faithful (the cocompact case is due to Guichard). This applies in particular to a version of Johnson-Millson bending deformations, providing for all $n$ infnitely many non-cocompact lattices in $\mathrm{SO}(n,1)$ which admit discrete and faithful deformations into $\mathrm{SU}(n,1)$. We also produce deformations of the figure-8 knot group into $\mathrm{SU}(3,1)$, not of bending type, to which the result applies.

There is a gap in the proof of Proposition 2.2, impacting the main results (Theorems 1.2, 1.4, 1.5). Sections 3 and 4 are not impacted except for Corollary 3.3

References in corpus (2)

• Complex hyperbolic triangle groups
• Bending deformations of complex hyperbolic surfaces