Deformation space of discrete groups of SU(2,1) in quaternionic hyperbolic plane

TL;DR

This paper investigates how certain discrete groups of SU(2,1) deform within quaternionic hyperbolic space, revealing contrasting behaviors in specific 3-manifold group representations.

Contribution

It provides the first analysis of deformation spaces of SU(2,1) groups in quaternionic hyperbolic space with explicit examples from 3-manifold groups.

Findings

One group is rigid outside U(2,1)

Another group admits extensive deformations in Sp(2,1)

Different behaviors depend on the specific 3-manifold group representations

Abstract

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