Group invariant Peano curves

Cannon, James W¹ ; Thurston, William P²

¹ 279 TMCB, Brigham Young University, Provo, UT 84602, USA
² Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA

Geometry & topology, Tome 11 (2007) no. 3, pp. 1315-1355.

Résumé

Our main theorem is that, if $M$ is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre $S$ and pseudo-Anosov monodromy, then the lift of the inclusion of $S$ in $M$ to universal covers extends to a continuous map of $B^2$ to $B^3$, where $B^n=H^n\cup S_{\infty }^{n-1}$. The restriction to $S_{\infty }^1$ maps onto $S_{\infty }^2$ and gives an example of an equivariant $S^2$–filling Peano curve. After proving the main theorem, we discuss the case of the figure-eight knot complement, which provides evidence for the conjecture that the theorem extends to the case when $S$ is a once-punctured hyperbolic surface.

Détail

DOI : 10.2140/gt.2007.11.1315

Keywords: Peano curve, group invariance, hyperbolic structure, 3–manifold, pseudo-Anosov diffeomorphism, fiber bundle over $S^1$

Bibliographie

[1] W Abikoff, Two theorems on totally degenerate Kleinian groups, Amer. J. Math. 98 (1976) 109

[2] F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. $(2)$ 124 (1986) 71

[3] J W Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984) 123

[4] A Fathi, F Laudenbach, V Poenaru, E Al, Travaux de Thurston sur les surfaces, Séminaire Orsay, Astérisque 66–67, Société Mathématique de France (1979) 1

[5] R L Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925) 416

[6] D Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $3$ fibrées sur $S^{1}$, from: "Séminaire Bourbaki 554 (1980)", Lecture Notes in Math. 842, Springer (1981) 196

[7] W P Thurston, Hyperbolic Structures on 3-manifolds, II: Surface groups and 3–manifolds which fiber over the circle

[8] W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997)