Canonical splittings of groups and 3manifolds
Authors:
Peter Scott and Gadde A. Swarup
Journal:
Trans. Amer. Math. Soc. 353 (2001), 49735001
MSC (2000):
Primary 57M07, 57N10, 20E06
Published electronically:
July 25, 2001
MathSciNet review:
1852090
Fulltext PDF Free Access
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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 JSJdecomposition of . This leads to a new proof of Johannson's Deformation Theorem.
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 1.
 B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145186. MR 99g:20069
 2.
 M. J. Dunwoody and M. Sageev, JSJ splittings for finitely presented groups over slender groups, Invent. Math. 135(1999), 2544. MR 2000b:20050
 3.
 B. Evans, Boundary respecting maps of manifolds, Pacific J. Math. 42 (1972), 639655. MR 47:9626
 4.
 M. H. Freedman, J. Hass and P. Scott, Least area incompressible surfaces in manifolds, Invent. Math. 71 (1983), 609642. MR 85e:57012
 5.
 F. Fujiwara and P. Papasoglu, JSJDecompositions of finitely presented groups and complexes of groups, Preprint (1998).
 6.
 W. Heil, On irreducible manifolds, Bull. Amer. Math. Soc. 75(1969), 772775. MR 40:4958
 7.
 W. Jaco, Lectures on threemanifold 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, 493524. 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, SpringerVerlag, 1979. MR 82c:57005
 11.
 P. K. Kim and J. L. Tollefson, PL involutions of fibered manifolds, Trans. Amer. Math. Soc. 232 (1977), 221237. MR 56:13223
 12.
 W. H. Meeks and P. Scott, Finite group actions on manifolds, Invent. Math. 86 (1986), 287346. 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), 409442. MR 84g:53016
 14.
 B. Leeb and P. Scott, A geometric characteristic splitting in all dimensions, Comm. Math. Helv. 75 (2000), 201215. CMP 2000:16
 15.
 N. Nakauchi, On free boundary Plateau problem for generaldimensional surfaces, Osaka J. Math. 21 (1984), 831841. MR 86i:49041
 16.
 W. D. Neumann and G.A. Swarup, Canonical decompositions of manifolds, Geometry and Topology, 1 (1997), 2140. 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), 53109. MR 98m:20044
 18.
 R. Schoen and S.T. Yau, Existence of incompressible minimal surfaces and the topology of threedimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127142. MR 81k:58029
 19.
 P. Scott, On sufficiently large 3manifolds, Quart. J. Math. Oxford Ser. (2) 23 (1972), 159172; correction, ibid. (2) 24 (1973), 527529. MR 52:4295
 20.
 P. Scott, A new proof of the annulus and torus theorems, American J. of Math. 102 (1980), 241277. 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), 485506. MR 86i:57016
 22.
 P. Scott, The symmetry of intersection numbers in group theory, Geometry and Topology 2(1998), 1129, 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), 179218. CMP 2000:16
 24.
 Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and rank 1 Lie groups, Geom. Funct. Anal. 7 (1997), 561593. MR 98j:20044
 25.
 G. A. Swarup, Boundary preserving maps of manifolds, Proc. Amer. Math. Soc. 78 (1980), no. 2, 291294. MR 81g:57008
 26.
 G. A. Swarup, On a theorem of Johannson, J. London Math. Soc. (2) (1978), no. 3, 560562. MR 80c:57007
 27.
 T. W. Tucker, Boundaryreducible manifolds and Waldhausen's theorem, Michigan Math. J. 20 (1973), 321327. 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, HercegNovi, Yugoslavia, Beograd 1969, 331332.
 29.
 F. Waldhausen, On irreducible manifolds which are sufficiently large, Ann. of Math., 87 (1968), 5688. 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/S0002994701028719
PII:
S 00029947(01)028719
Keywords:
3manifold,
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
Published electronically:
July 25, 2001
Additional Notes:
The first author was partially supported by NSF grant DMS 034681.
Article copyright:
© Copyright 2001
American Mathematical Society
