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The deformation space of non-orientable hyperbolic 3-manifolds

Juan Luis Durán Batalla, Joan Porti

Abstract

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $Δ$, we compute the deformation space of the pair $(M^3, Δ)$ (its Neumann Zagier parameter space). We also determine the variety of representations of $π_1(M^3)$ in $\mathrm{Isom}(\mathbb{H}^3)$ in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair $(M^3, Δ)$. We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.

The deformation space of non-orientable hyperbolic 3-manifolds

Abstract

We consider non-orientable hyperbolic 3-manifolds of finite volume . When has an ideal triangulation , we compute the deformation space of the pair (its Neumann Zagier parameter space). We also determine the variety of representations of in in a neighborhood of the holonomy. As a consequence, when some ends are non-orientable, there are deformations from the variety of representations that cannot be realized as deformations of the pair . We also discuss the metric completion of these structures and we illustrate the results on the Gieseking manifold.

Paper Structure

This paper contains 16 sections, 31 theorems, 88 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Deformation space from ideal triangulations
  4. Varieties of representations
  5. Orientation covering and the involution on representations
  6. Representations in $\mathrm{PSL}(2,\mathbb C)$
  7. Local coordinates
  8. Representations of the Klein bottle
  9. Metric completion
  10. Conifolds and cylindrical coordinates
  11. Conifolds bounded by a Klein bottle
  12. The radial structure
  13. Example: The Gieseking manifold
  14. The deformation space $\mathrm{Def}(M, \Delta)$
  15. The Gieseking manifold as a punctured torus bundle
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M^3$ be a complete non-orientable hyperbolic $3$-manifold of finite volume with a single end, that is non-orientable. Furthermore, the holonomy map $\mathrm{Def}(M^3, \Delta)\to \mathcal{R}(\pi_1(M^3),\mathrm{Isom}(\mathbb{H}^3))$ folds the interval $(-1,1)$ at 0 and its image is the half-open interval $[0,1)$, where $0$ corresponds to the complete structure.

Figures (12)

  • Figure 1: Change under the action of $\iota$.
  • Figure 2: Cylindrical coordinates.
  • Figure 3: Orthogonal projection to $\partial_\infty\mathbb H^3$ with $g$ the geodesic with ideal end-points $0$ and $\infty$.
  • Figure 4: A solid torus as two $3$-balls joined by two $1$-handles.
  • Figure 5: The radial thickening.
  • ...and 7 more figures

Theorems & Definitions (76)

  • Theorem 1.1
  • Theorem 2.1: Neumann--Zagier NeumannZagier
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • ...and 66 more