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Canonical splittings of groups and 3-manifolds
Authors:
Peter Scott and Gadde A. Swarup
Journal:
Trans. Amer. Math. Soc. 353 (2001), 4973-5001
MSC (2000):
Primary 57M07, 57N10, 20E06
Posted:
July 25, 2001
MathSciNet review:
1852090
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Abstract: We introduce the notion of a `canonical' splitting over  or  for a finitely generated group  . We show that when  happens to be the fundamental group of an orientable Haken manifold  with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related to the one given by the standard JSJ-decomposition of  . This leads to a new proof of Johannson's Deformation Theorem.
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- 20.
Peter
Scott, A new proof of the annulus and torus theorems, Amer. J.
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Peter
Scott, Strong annulus and torus theorems and the enclosing property
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- 22.
Peter
Scott, The symmetry of intersection numbers in group theory,
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Peter
Scott, Correction to: “The symmetry of intersection numbers
in group theory”, Geom. Topol. 2 (1998),
Erratum 1 (electronic). MR 1639541
(99k:20076b), http://dx.doi.org/10.2140/gt.1998.2.11
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P. Scott and G. A. Swarup, Splittings of groups and intersection numbers, Geometry and Topology 4 (2000), 179-218. CMP 2000:16
- 24.
Z.
Sela, Structure and rigidity in (Gromov) hyperbolic groups and
discrete groups in rank 1 Lie groups. II, Geom. Funct. Anal.
7 (1997), no. 3, 561–593. MR 1466338
(98j:20044), http://dx.doi.org/10.1007/s000390050019
- 25.
G.
A. Swarup, Boundary preserving maps of
3-manifolds, Proc. Amer. Math. Soc.
78 (1980), no. 2,
291–294. MR
550516 (81g:57008), http://dx.doi.org/10.1090/S0002-9939-1980-0550516-X
- 26.
G.
A. Swarup, On a theorem of Johannson, J. London Math. Soc. (2)
18 (1978), no. 3, 560–562. MR 518243
(80c:57007), http://dx.doi.org/10.1112/jlms/s2-18.3.560
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T.
W. Tucker, Boundary-reducible 3-manifolds and Waldhausen’s
theorem, Michigan Math. J. 20 (1973), 321–327.
MR
0334218 (48 #12537)
- 28.
F. Waldhausen, On the determination of some bounded
 -manifolds by their fundamental group alone, Proc. of the Internat. Symp. on Topology and its Applications, Herceg-Novi, Yugoslavia, Beograd 1969, 331-332.
- 29.
Friedhelm
Waldhausen, Eine Verallgemeinerung des Schleifensatzes,
Topology 6 (1967), 501–504 (German). MR 0220300
(36 #3366)
- 1.
- B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145-186. MR 99g:20069
- 2.
- M. J. Dunwoody and M. Sageev, JSJ splittings for finitely presented groups over slender groups, Invent. Math. 135(1999), 25-44. MR 2000b:20050
- 3.
- B. Evans, Boundary respecting maps of
 -manifolds, Pacific J. Math. 42 (1972), 639-655. MR 47:9626
- 4.
- M. H. Freedman, J. Hass and P. Scott, Least area incompressible surfaces in
 -manifolds, Invent. Math. 71 (1983), 609-642. MR 85e:57012
- 5.
- F. Fujiwara and P. Papasoglu, JSJ-Decompositions of finitely presented groups and complexes of groups, Preprint (1998).
- 6.
- W. Heil, On
 -irreducible  -manifolds, Bull. Amer. Math. Soc. 75(1969), 772-775. MR 40:4958
- 7.
- W. Jaco, Lectures on three-manifold topology, Amer. Math. Soc., Providence, R.I., 1980. MR 81k:57009
- 8.
- W. Jaco and J. H. Rubinstein, PL minimal surfaces in
 -manifolds, J. Differential Geom. 27(1988), no.3, 493-524. MR 89e:57009
- 9.
- W. Jaco and P. B. Shalen, Seifert fibered spaces in
 -manifolds, Memoirs of Amer. Math. Soc., vol. 21, Number 220 (1979). MR 81c:57010
- 10.
- K. Johannson, Homotopy equivalences of
 -manifolds with boundaries, Lecture Notes in Mathematics 761, Springer-Verlag, 1979. MR 82c:57005
- 11.
- P. K. Kim and J. L. Tollefson, PL involutions of fibered
 -manifolds, Trans. Amer. Math. Soc. 232 (1977), 221-237. MR 56:13223
- 12.
- W. H. Meeks and P. Scott, Finite group actions on
 -manifolds, Invent. Math. 86 (1986), 287-346. MR 88b:57039
- 13.
- W. H. Meeks, III and S.-T. Yau, The classical Plateau problem and the topology of three dimensional manifolds, Topology 21 (1982), 409-442. MR 84g:53016
- 14.
- B. Leeb and P. Scott, A geometric characteristic splitting in all dimensions, Comm. Math. Helv. 75 (2000), 201-215. CMP 2000:16
- 15.
- N. Nakauchi, On free boundary Plateau problem for general-dimensional surfaces, Osaka J. Math. 21 (1984), 831-841. MR 86i:49041
- 16.
- W. D. Neumann and G.A. Swarup, Canonical decompositions of
 -manifolds, Geometry and Topology, 1 (1997), 21-40. MR 98k:57033
- 17.
- E. Rips and Z. Sela, Cyclic splittings of finitely generated groups and the canonical JSJ decomposition, Ann. of Math. 146 (1997), 53-109. MR 98m:20044
- 18.
- R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127-142. MR 81k:58029
- 19.
- P. Scott, On sufficiently large 3-manifolds, Quart. J. Math. Oxford Ser. (2) 23 (1972), 159-172; correction, ibid. (2) 24 (1973), 527-529. MR 52:4295
- 20.
- P. Scott, A new proof of the annulus and torus theorems, American J. of Math. 102 (1980), 241-277. MR 81f:57006
- 21.
- P. Scott, Strong annulus and torus theorems and the enclosing property of characteristic submanifolds, Quarterly J. of Math. Oxford (2), 35 (1984), 485-506. MR 86i:57016
- 22.
- P. Scott, The symmetry of intersection numbers in group theory, Geometry and Topology 2(1998), 11-29, Correction (ibid) (1998). MR 99k:20076a;MR 99k:20076b
- 23.
- P. Scott and G. A. Swarup, Splittings of groups and intersection numbers, Geometry and Topology 4 (2000), 179-218. CMP 2000:16
- 24.
- Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and rank 1 Lie groups, Geom. Funct. Anal. 7 (1997), 561-593. MR 98j:20044
- 25.
- G. A. Swarup, Boundary preserving maps of
 -manifolds, Proc. Amer. Math. Soc. 78 (1980), no. 2, 291-294. MR 81g:57008
- 26.
- G. A. Swarup, On a theorem of Johannson, J. London Math. Soc. (2) (1978), no. 3, 560-562. MR 80c:57007
- 27.
- T. W. Tucker, Boundary-reducible
 -manifolds and Waldhausen's theorem, Michigan Math. J. 20 (1973), 321-327. MR 48:12537
- 28.
- F. Waldhausen, On the determination of some bounded -manifolds by their fundamental group alone, Proc. of the Internat. Symp. on Topology and its Applications, Herceg-Novi, Yugoslavia, Beograd 1969, 331-332.
- 29.
- F. Waldhausen, On irreducible
 -manifolds which are sufficiently large, Ann. of Math., 87 (1968), 56-88. MR 36:3366
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Additional Information
Peter Scott
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email:
pscott@math.lsa.umich.edu
Gadde A. Swarup
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
Email:
gadde@ms.unimelb.edu.au
DOI:
http://dx.doi.org/10.1090/S0002-9947-01-02871-9
PII:
S 0002-9947(01)02871-9
Keywords:
3-manifold,
characteristic submanifold,
Deformation Theorem,
ends of groups,
intersection number,
JSJ decomposition,
splittings of groups
Received by editor(s):
August 12, 2000
Received by editor(s) in revised form:
April 9, 2001
Posted:
July 25, 2001
Additional Notes:
The first author was partially supported by NSF grant DMS 034681.
Article copyright:
© Copyright 2001 American Mathematical Society
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