Mapping class groups of medium distance Heegaard splittings

Johnson, Jesse

doi:10.1090/S0002-9939-2010-10545-2

Mapping class groups of medium distance Heegaard splittings
HTML articles powered by AMS MathViewer

by Jesse Johnson PDF

Proc. Amer. Math. Soc. 138 (2010), 4529-4535 Request permission

Abstract:

We show that if the Hempel distance of a Heegaard splitting is larger than three, then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that given two handlebody sets in the curve complex for a surface that are distance at least four apart, the group of automorphisms of the curve complex that preserve both handlebody sets is finite.

References

John Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001), no. 3, 631–657. MR 1838999, DOI 10.1016/S0040-9383(00)00033-1
Nikolai V. Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices 14 (1997), 651–666. MR 1460387, DOI 10.1155/S1073792897000433
J. Johnson, Flipping and stabilizing Heegaard splittings, preprint (2008), arXiv:0805.4422.
J. Johnson and H. Rubinstein, Mapping class groups of Heegaard splittings, preprint (2006), arXiv:math.GT/0701119.
Tsuyoshi Kobayashi and Osamu Saeki, The Rubinstein-Scharlemann graphic of a 3-manifold as the discriminant set of a stable map, Pacific J. Math. 195 (2000), no. 1, 101–156. MR 1781617, DOI 10.2140/pjm.2000.195.101
D. D. Long, On pseudo-Anosov maps which extend over two handlebodies, Proc. Edinburgh Math. Soc. (2) 33 (1990), no. 2, 181–190. MR 1057747, DOI 10.1017/S0013091500018113
Feng Luo, Automorphisms of the complex of curves, Topology 39 (2000), no. 2, 283–298. MR 1722024, DOI 10.1016/S0040-9383(99)00008-7
Hossein Namazi, Big Heegaard distance implies finite mapping class group, Topology Appl. 154 (2007), no. 16, 2939–2949. MR 2355879, DOI 10.1016/j.topol.2007.05.011
Hyam Rubinstein and Martin Scharlemann, Comparing Heegaard splittings of non-Haken $3$-manifolds, Topology 35 (1996), no. 4, 1005–1026. MR 1404921, DOI 10.1016/0040-9383(95)00055-0
Martin Scharlemann and Maggy Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10 (2006), 593–617. MR 2224466, DOI 10.2140/gt.2006.10.593
Abigail Thompson, The disjoint curve property and genus $2$ manifolds, Topology Appl. 97 (1999), no. 3, 273–279. MR 1711418, DOI 10.1016/S0166-8641(98)00063-7