Rank 1 deformations of non-cocompact hyperbolic lattices

TL;DR

This paper proves that small, parabolic-preserving deformations of non-cocompact lattices in negatively curved symmetric spaces remain discrete and faithful, extending known results and providing new examples of such deformations.

Contribution

It extends the theory of deformations of non-cocompact lattices, showing they remain discrete and faithful under certain conditions, and constructs new examples into SU(n,1).

Findings

Small deformations preserve discreteness and faithfulness.

Infinitely many non-cocompact lattices admit deformations into SU(n,1).

Constructed deformations of the figure-8 knot group into SU(3,1).

Abstract

Let $X$ be a negatively curved symmetric space and $\Gamma$ a non-cocompact lattice in $\rm{Isom}(X)$. 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 ${\rm SO}(n,1)$ which admit discrete and faithful deformations into ${\rm SU}(n,1)$. We also produce deformations of the figure-8 knot group into $\rm{SU}(3,1)$, not of bending type, to which the result applies.