Local rigidity for PGL(3,C)-representations of 3-manifold groups

TL;DR

This paper develops local coordinates for PGL(3,C)-representation varieties of hyperbolic 3-manifolds and proves local rigidity of the geometric representation, extending previous results and analyzing specific examples.

Contribution

It introduces a method to produce local coordinates for PGL(3,C) representation varieties and establishes criteria for local rigidity, including detailed analysis of specific manifolds.

Findings

Proved local rigidity of the geometric PGL(3,C) representation.

Provided a criterion for local rigidity of PGL(3,C) representations.

Analyzed the figure eight knot sister manifold to illustrate different rigidity scenarios.

Abstract

Let M be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetraedra. We explain how to produce local coordinates for the variety defined by the gluing equations for PGL(3,C)-representations. In particular we prove local rigidity of the "geometric" representation in PGL(3,C), recovering a recent result of Menal-Ferrer and Porti. More generally we give a criterion for local rigidty of PGL(3,C)-representations and provide detailed analysis of the figure eight knot sister manifold exhibiting the different possibilities that can occur.