A generalisation of the deformation variety

Given an ideal tetrahedralisation of a connected 3-manifold with nonempty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to the gluing equations, from which one can recover a representation of the fundamental group of the manifold into the isometries of 3-dimensional hyperbolic space. However, the deformation variety depends crucially on the tetrahedralisation: there may be entire components of the representation variety which can be obtained from the deformation variety with one tetrahedralisation but not another. We introduce a generalisation of the deformation variety, which again consists of assignments of complex variables to certain dihedral angles subject to polynomial equations, but together with some extra combinatorial data concerning degenerate tetrahedra. This extended deformation variety deals with many situations that the deformation variety cannot. In particular, we show that given a manifold that admits an ideal tetrahedralisation, we can construct a tetrahedralisation so that we can recover any irreducible representation whose image is not a generalised dihedral group from the associated extended deformation variety. This paper is organised as follows: in section 1, we recall the de nition of the deformation variety and give an example showing how a bad choice of tetrahedralisation can cause problems. In section 2, we introduce Serre's tree as a better context to work in and prove some lemmas about developing cross ratios in the tree. In section 3, we de ne the extended deformation variety and its developing map, and show that it is an a ne algebraic variety. In section 4 show how to alter a tetrahedralisation so that it has properties needed for the extended deformation variety. In section 5, we de ne the map from the extended deformation variety to the representation variety and prove (in theorem 5.11) that under good conditions (i.e. a good tetrahedralisation, which we can ensure using the results of section 4), this map hits all of the representations that are irreducible and do not have as image in PSL2(C) a generalised dihedral group. In section 6, we return to the example of section 1 and show how the extended deformation variety deals with it, and in section 7, we ask some questions and suggest further directions to explore. The author thanks Eric Katz and Alan Reid for helpful discussions. 1 ar X iv :0 90 4. 18 93 v1 [ m at h. G T ] 1 3 A pr 2 00 9 1 The deformation variety De nition 1.1. Let M be a 3-manifold with non-empty boundary consisting of a disjoint union of tori, and with ideal tetrahedralisation T consisting of N tetrahedra. The deformation variety of M with respect to the tetrahedralisation T, D(M) = D(M ; T) is the a ne variety in (C \ {0, 1}) , de ned as the solutions of gluing equations and identities between complex dihedral angles within each tetrahedron, where each of the three complex dihedral angles in each tetrahedron corresponds to a dimension of the ambient space. Speci cally, for the 6 dihedral angles within each tetrahedron, angles on opposite edges are the same, as shown in gure 1 (hence the fact that there are three complex variables for each of the N tetrahedra), and x1, x2, x3 are related to each other by: x1x2x3 = −1 (1) x1x2 − x1 + 1 = 0 (2) For each edge of T, we also require that the product of the complex dihedral angles arranged around an edge of the tetrahedralisation equals 1 (these are the gluing equations). This de nition is essentially rst seen in Thurston's notes [3], chapter 4.