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Regular neighbourhoods and canonical decompositions for groups
Author(s):
Peter
Scott;
Gadde
A.
Swarup
Journal:
Electron. Res. Announc. Amer. Math. Soc.
8
(2002),
20-28.
MSC (2000):
Primary 20E34;
Secondary 57N10, 57M07
Posted:
September 6, 2002
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Abstract:
We find canonical decompositions for finitely presented groups which essentially specialise to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood of a family of almost invariant subsets of a group. An almost invariant set is an analogue of an immersion.
References:
- 1.
- M. Bestvina and M. Feighn, Bounding the complexity of simplicial actions on trees, Invent. Math. 103 (1991), 145-186. MR 92c:20044
- 2.
- B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), 145-186. MR 99g:20069
- 3.
- B. H. Bowditch, Splittings of finitely generated groups over two-ended subgroups, Trans. Amer. Math. Soc. 354 (2002), no. 3, 1049-1078.
- 4.
- M. J. Dunwoody and M. Roller, Splitting groups over polycyclic-by-finite subgroups, Bull. London Math. Soc. 25 (1993), 29-36. MR 93h:20029
- 5.
- M. J. Dunwoody and M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), 25-44. MR 2000b:20050
- 6.
- M. J. Dunwoody and E. Swenson, The Algebraic Torus Theorem, Invent. Math. 140 (2000), 605-637. MR 2001d:20039
- 7.
- M. Freedman, J. Hass, and G. P. Scott, Least area incompressible surfaces in
 -manifolds, Invent. Math. 71 (1983), 609-642. MR 85e:57012 - 8.
- K. Fujiwara and P. Papasoglu, JSJ-decompositions for finitely presented groups and complexes of groups, Preprint (1997).
- 9.
- W. Jaco and P. 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 boundary, Lecture Notes in Mathematics, vol. 761, Springer-Verlag, 1979. MR 82c:57005 - 11.
- P. H. Kropholler, An analogue of the torus decomposition theorem for certain Poincaré duality groups, Proc. London. Math. Soc. (3) 60 (1990), 503-529. MR 91g:20079
- 12.
- G. Mess, Examples of Poincaré duality groups, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1145-1146. MR 91c:20075
- 13.
- E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and canonical JSJ-decomposition, Annals of Mathematics 146 (1997), 53-109. MR 98m:20044
- 14.
- 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
- 15.
- P. Scott and G. A. Swarup, Splittings of groups and intersection numbers, Geometry and Topology 4 (2000), 179-218. MR 2001h:20032
- 16.
- P. Scott and G. A. Swarup, An Algebraic Annulus Theorem, Pacific Journal of Math. 196 (2000), 461-506. MR 2001k:20090
- 17.
- P. Scott and G. A. Swarup, Canonical splittings of groups and
 -manifolds, Trans. Amer. Math. Soc. 353 (2001), 4973-5001. MR 2002f:57002 - 18.
- Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and rank 1 Lie groups, Geometric and Functional Anal. 7 (1997), 561-593. MR 98j:20044
- 19.
- F. Waldhausen, On the determination of some bounded
 -manifolds by their fundamental groups alone, Proc. of the International Sympo. on Topology and Its Applications, Herceg-Novi, Yugoslavia, Beograd, 1969, 331-332. MR 42:2416
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Additional Information:
Peter
Scott
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA
Email:
pscott@umich.edu
Gadde
A.
Swarup
Affiliation:
Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
Email:
gadde@ms.unimelb.edu.au
DOI:
10.1090/S1079-6762-02-00102-6
PII:
S 1079-6762(02)00102-6
Keywords:
Graph of groups,
almost invariant set,
characteristic submanifold
Received by editor(s):
May 1, 2002,
Received by editor(s) in revised form:
July 23, 2002
Posted:
September 6, 2002
Additional Notes:
First author partially supported by NSF grants DMS 034681 and 9626537
Communicated by:
Walter Neumann
Copyright of article:
Copyright
2002,
American Mathematical Society
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