Local Rigidity for -Representations of 3-Manifold Groups

Abstract

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

2000 AMS Subject Classification:

• 57N10
• 57M27
• 11R70
• 19F27
• 11G55
• 68W30
• 14Q99

Keywords:

• rigidity of representations
• computational aspects in algebraic geometry
• PGL(3, )
• 3-manifolds
• flags

Acknowledgments

This research was financed in part by the ANR project Structures Géométriques Triangulées. N. B. is a member of Institut Universitaire de France.

Notes

¹Note, however, that starting from the Epstein–Penner decomposition of M into ideal polyhedra, [Petronio and Porti 00] produces a degenerate triangulation of M.

²This product should be interpreted as the Killing form on the space of roots of through a suitable choice of basis.