Deformation space of a non-uniform 3-dimensional real hyperbolic lattice in quaternionic hyperbolic plane

TL;DR

This paper investigates the deformation space of a specific non-uniform hyperbolic lattice in quaternionic hyperbolic space, showing rigidity of certain representations related to the figure eight knot complement.

Contribution

It demonstrates the rigidity of the fundamental group representations of the figure eight knot complement in quaternionic hyperbolic space, extending understanding of deformation spaces in this context.

Findings

Representations into PU(2,1) cannot be deformed in PSp(2,1) outside PU(2,1)

Rigidity result for the figure eight knot complement in quaternionic hyperbolic space

Insights into deformation spaces of hyperbolic lattices

Abstract

In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in $PSp(2,1)$ out of PU(2,1) up to conjugacy.