Mapping class groups of medium distance Heegaard splittings

Author:
Jesse Johnson

Journal:
Proc. Amer. Math. Soc. **138** (2010), 4529-4535

MSC (2010):
Primary 57Mxx

Published electronically:
July 20, 2010

MathSciNet review:
2680077

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Abstract | References | Similar Articles | Additional Information

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.

**1.**John Hempel,*3-manifolds as viewed from the curve complex*, Topology**40**(2001), no. 3, 631–657. MR**1838999**, 10.1016/S0040-9383(00)00033-1**2.**Nikolai V. Ivanov,*Automorphism of complexes of curves and of Teichmüller spaces*, Internat. Math. Res. Notices**14**(1997), 651–666. MR**1460387**, 10.1155/S1073792897000433**3.**J. Johnson,*Flipping and stabilizing Heegaard splittings*, preprint (2008), arXiv:0805.4422.**4.**J. Johnson and H. Rubinstein,*Mapping class groups of Heegaard splittings*, preprint (2006), arXiv:math.GT/0701119.**5.**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**, 10.2140/pjm.2000.195.101**6.**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**, 10.1017/S0013091500018113**7.**Feng Luo,*Automorphisms of the complex of curves*, Topology**39**(2000), no. 2, 283–298. MR**1722024**, 10.1016/S0040-9383(99)00008-7**8.**Hossein Namazi,*Big Heegaard distance implies finite mapping class group*, Topology Appl.**154**(2007), no. 16, 2939–2949. MR**2355879**, 10.1016/j.topol.2007.05.011**9.**Hyam Rubinstein and Martin Scharlemann,*Comparing Heegaard splittings of non-Haken 3-manifolds*, Topology**35**(1996), no. 4, 1005–1026. MR**1404921**, 10.1016/0040-9383(95)00055-0**10.**Martin Scharlemann and Maggy Tomova,*Alternate Heegaard genus bounds distance*, Geom. Topol.**10**(2006), 593–617 (electronic). MR**2224466**, 10.2140/gt.2006.10.593**11.**Abigail Thompson,*The disjoint curve property and genus 2 manifolds*, Topology Appl.**97**(1999), no. 3, 273–279. MR**1711418**, 10.1016/S0166-8641(98)00063-7

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Additional Information

**Jesse Johnson**

Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078

Email:
jjohnson@math.okstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10545-2

Keywords:
Heegaard splitting,
mapping class group,
curve complex

Received by editor(s):
November 16, 2009

Published electronically:
July 20, 2010

Additional Notes:
This research was supported by NSF MSPRF grant 0602368

Communicated by:
Daniel Ruberman

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.