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
DOI:
https://doi.org/10.1090/S1079-6762-02-00102-6
Published electronically:
September 6, 2002
MathSciNet review:
1928498
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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.
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B1B. H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), 145–186.
B2B. H. Bowditch, Splittings of finitely generated groups over two-ended subgroups, Trans. Amer. Math. Soc. 354 (2002), no. 3, 1049–1078.
D-RollerM. J. Dunwoody and M. Roller, Splitting groups over polycyclic-by-finite subgroups, Bull. London Math. Soc. 25 (1993), 29–36.
D-SageevM. J. Dunwoody and M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), 25–44.
D-SwensonM. J. Dunwoody and E. Swenson, The Algebraic Torus Theorem, Invent. Math. 140 (2000), 605–637.
FHSM. Freedman, J. Hass, and G. P. Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), 609–642.
FPK. Fujiwara and P. Papasoglu, JSJ-decompositions for finitely presented groups and complexes of groups, Preprint (1997).
JSW. Jaco and P. Shalen, Seifert fibered spaces in $3$-manifolds, Memoirs of Amer. Math. Soc. vol. 21, Number 220 (1979).
JOK. Johannson, Homotopy equivalences of $3$-manifolds with boundary, Lecture Notes in Mathematics, vol. 761, Springer-Verlag, 1979.
KP. H. Kropholler, An analogue of the torus decomposition theorem for certain Poincaré duality groups, Proc. London. Math. Soc. (3) 60 (1990), 503–529.
MessG. Mess, Examples of Poincaré duality groups, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1145–1146.
RSE. Rips and Z. Sela, Cyclic splittings of finitely presented groups and canonical JSJ-decomposition, Annals of Mathematics 146 (1997), 53–109.
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SS3P. Scott and G. A. Swarup, Canonical splittings of groups and $3$-manifolds, Trans. Amer. Math. Soc. 353 (2001), 4973–5001.
S1Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and rank 1 Lie groups, Geometric and Functional Anal. 7 (1997), 561–593.
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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
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
