Deformation Spaces of Discrete Groups of SU(2,1) in Quaternionic Hyperbolic Plane: A Case Study

Abstract

In this note, we study deformations of discrete and Zariski dense subgroups of $\text{SU}(2,1)$ in the isometry group $\text{Sp}(2,1)$ of quaternionic hyperbolic space. Specifically, we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behaviors: one is not deformable outside U(2, 1), while the other has a big space of deformations in $\text{Sp}(2,1)$.

Keywords:

• Quaternionic hyperbolic space
• complex hyperbolic space
• local rigidity
• representation variety

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 51M10
• 57S25

Notes

2 Even though ρ₀ is a boundary unipotent representation in $\text{PU}(2,1)$, the cohomology $H^1({\Gamma }_8,S^2{ℂ}^3)$ vanishes contrary to the case $H^1({\Gamma }_8,S^2{ℂ}^2)$ of a boundary unipotent representation. This is not a contradiction, as we are considering $S^2{ℂ}^3$ and the representation is not boundary unipotent in $\mathrm{U}(2,1)$.

Additional information

Funding

The second author gratefully acknowledges the partial support of the grant (NRF-2017R1A2A2A05001002) and a warm support of IHES during his stay.