Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



Regular neighbourhoods and canonical decompositions for groups

Authors: Peter Scott and Gadde A. Swarup
Journal: Electron. Res. Announc. Amer. Math. Soc. 8 (2002), 20-28
MSC (2000): Primary 20E34; Secondary 57N10, 57M07
Published electronically: September 6, 2002
MathSciNet review: 1928498
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 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 $3$-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 $3$-manifolds, Memoirs of Amer. Math. Soc. vol. 21, Number 220 (1979). MR 81c:57010
  • 10. K. Johannson, Homotopy equivalences of $3$-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 $3$-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 $3$-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

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 20E34, 57N10, 57M07

Retrieve articles in all journals with MSC (2000): 20E34, 57N10, 57M07

Additional Information

Peter Scott
Affiliation: Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA

Gadde A. Swarup
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

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
Published electronically: September 6, 2002
Additional Notes: First author partially supported by NSF grants DMS 034681 and 9626537
Communicated by: Walter Neumann
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society