A Cantor set with hyperbolic complement

Souto, Juan; Stover, Matthew

doi:10.1090/S1088-4173-2013-00249-X

A Cantor set with hyperbolic complement
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by Juan Souto and Matthew Stover

Conform. Geom. Dyn. 17 (2013), 58-67

DOI: https://doi.org/10.1090/S1088-4173-2013-00249-X

Published electronically: April 22, 2013

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Abstract:

We construct a Cantor set in $\mathbb {S}^3$ whose complement admits a complete hyperbolic metric.

References

I. Agol, Tameness of hyperbolic $3$-manifolds, math.GT/0405568.
M. L. Antoine, Sur la possibilité d’étendre l’homéiomorphie de deux figures à leur voisinages, C.R. Acad. Sci. Paris 171 (1920), 661–664.
M. Bestvina and D. Cooper, A wild Cantor set as the limit set of a conformal group action on $S^3$, Proc. Amer. Math. Soc. 99 (1987), no. 4, 623–626. MR 877028, DOI 10.1090/S0002-9939-1987-0877028-4
R. H. Bing, A homeomorphism between the $3$-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354–362. MR 49549, DOI 10.2307/1969804
Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446. MR 2188131, DOI 10.1090/S0894-0347-05-00513-8
D. G. DeGryse and R. P. Osborne, A wild Cantor set in $E^{n}$ with simply connected complement, Fund. Math. 86 (1974), 9–27. MR 375323, DOI 10.4064/fm-86-1-9-27
Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
Michael H. Freedman and Richard Skora, Strange actions of groups on spheres, J. Differential Geom. 25 (1987), no. 1, 75–98. MR 873456
Dennis J. Garity, Dušan Repovš, and Matjaž Željko, Rigid Cantor sets in $\textbf {R}^3$ with simply connected complement, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2447–2456. MR 2213719, DOI 10.1090/S0002-9939-06-08459-0
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744, DOI 10.1007/BFb0085406
T. Jørgensen and A. Marden, Algebraic and geometric convergence of Kleinian groups, Math. Scand. 66 (1990), no. 1, 47–72. MR 1060898, DOI 10.7146/math.scand.a-12292
Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613
R. Kent and J. Souto, Geometric limits of knot complements, II: Graphs determined by their complements, accepted for publication in Mathematische Zeitschrift.
A. Marden, Outer circles, Cambridge University Press, Cambridge, 2007. An introduction to hyperbolic 3-manifolds. MR 2355387, DOI 10.1017/CBO9780511618918
Robert Myers, Excellent $1$-manifolds in compact $3$-manifolds, Topology Appl. 49 (1993), no. 2, 115–127. MR 1206219, DOI 10.1016/0166-8641(93)90038-F
Jean-Pierre Otal, Thurston’s hyperbolization of Haken manifolds, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996) Int. Press, Boston, MA, 1998, pp. 77–194. MR 1677888
R. B. Sher, Concerning wild Cantor sets in $E^{3}$, Proc. Amer. Math. Soc. 19 (1968), 1195–1200. MR 234438, DOI 10.1090/S0002-9939-1968-0234438-4
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
Richard Skora, Cantor sets in $S^3$ with simply connected complements, Topology Appl. 24 (1986), no. 1-3, 181–188. Special volume in honor of R. H. Bing (1914–1986). MR 872489, DOI 10.1016/0166-8641(86)90060-X
William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594