Canonical splittings of groups and 3-manifolds

Scott, Peter; Swarup, Gadde

doi:10.1090/S0002-9947-01-02871-9

Canonical splittings of groups and 3-manifolds
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by Peter Scott and Gadde A. Swarup

Trans. Amer. Math. Soc. 353 (2001), 4973-5001

DOI: https://doi.org/10.1090/S0002-9947-01-02871-9

Published electronically: July 25, 2001

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Abstract:

We introduce the notion of a ‘canonical’ splitting over $\mathbb {Z}$ or $\mathbb {Z}\times \mathbb {Z}$ for a finitely generated group $G$. We show that when $G$ happens to be the fundamental group of an orientable Haken manifold $M$ 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 $M$. This leads to a new proof of Johannson’s Deformation Theorem.

References

Brian H. Bowditch, Cut points and canonical splittings of hyperbolic groups, Acta Math. 180 (1998), no. 2, 145–186. MR 1638764, DOI 10.1007/BF02392898
M. J. Dunwoody and M. E. Sageev, JSJ-splittings for finitely presented groups over slender groups, Invent. Math. 135 (1999), no. 1, 25–44. MR 1664694, DOI 10.1007/s002220050278
Benny Evans, Boundary respecting maps of $3$-mainfolds, Pacific J. Math. 42 (1972), 639–655. MR 321093
Michael Freedman, Joel Hass, and Peter Scott, Least area incompressible surfaces in $3$-manifolds, Invent. Math. 71 (1983), no. 3, 609–642. MR 695910, DOI 10.1007/BF02095997
F. Fujiwara and P. Papasoglu, JSJ-Decompositions of finitely presented groups and complexes of groups, Preprint (1998).
Wolfgang Heil, On $P^{2}$-irreducible $3$-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775. MR 251731, DOI 10.1090/S0002-9904-1969-12283-4
William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450
William Jaco and J. Hyam Rubinstein, PL minimal surfaces in $3$-manifolds, J. Differential Geom. 27 (1988), no. 3, 493–524. MR 940116
William H. Jaco and Peter B. Shalen, Seifert fibered spaces in $3$-manifolds, Mem. Amer. Math. Soc. 21 (1979), no. 220, viii+192. MR 539411, DOI 10.1090/memo/0220
Klaus Johannson, Homotopy equivalences of $3$-manifolds with boundaries, Lecture Notes in Mathematics, vol. 761, Springer, Berlin, 1979. MR 551744
Paik Kee Kim and Jeffrey L. Tollefson, PL involutions of fibered $3$-manifolds, Trans. Amer. Math. Soc. 232 (1977), 221–237. MR 454981, DOI 10.1090/S0002-9947-1977-0454981-2
William H. Meeks III and Peter Scott, Finite group actions on $3$-manifolds, Invent. Math. 86 (1986), no. 2, 287–346. MR 856847, DOI 10.1007/BF01389073
William H. Meeks III and Shing Tung Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn’s lemma, Topology 21 (1982), no. 4, 409–442. MR 670745, DOI 10.1016/0040-9383(82)90021-0
B. Leeb and P. Scott, A geometric characteristic splitting in all dimensions, Comm. Math. Helv. 75 (2000), 201-215.
Nobumitsu Nakauchi, On free boundary Plateau problem for general-dimensional surfaces, Osaka J. Math. 21 (1984), no. 4, 831–841. MR 765359
Walter D. Neumann and Gadde A. Swarup, Canonical decompositions of $3$-manifolds, Geom. Topol. 1 (1997), 21–40. MR 1469066, DOI 10.2140/gt.1997.1.21
E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Ann. of Math. (2) 146 (1997), no. 1, 53–109. MR 1469317, DOI 10.2307/2951832
R. Schoen and Shing Tung Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), no. 1, 127–142. MR 541332, DOI 10.2307/1971247
G. 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 383414, DOI 10.1093/qmath/23.2.159
Peter Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980), no. 2, 241–277. MR 564473, DOI 10.2307/2374238
Peter Scott, Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of $3$-manifolds, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 140, 485–506. MR 767777, DOI 10.1093/qmath/35.4.485
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
P. Scott and G. A. Swarup, Splittings of groups and intersection numbers, Geometry and Topology 4 (2000), 179-218.
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, DOI 10.1007/s000390050019
G. A. Swarup, Boundary preserving maps of $3$-manifolds, Proc. Amer. Math. Soc. 78 (1980), no. 2, 291–294. MR 550516, DOI 10.1090/S0002-9939-1980-0550516-X
G. A. Swarup, On a theorem of Johannson, J. London Math. Soc. (2) 18 (1978), no. 3, 560–562. MR 518243, DOI 10.1112/jlms/s2-18.3.560
T. W. Tucker, Boundary-reducible $3$-manifolds and Waldhausen’s theorem, Michigan Math. J. 20 (1973), 321–327. MR 334218
F. Waldhausen, On the determination of some bounded $3$-manifolds by their fundamental group alone, Proc. of the Internat. Symp. on Topology and its Applications, Herceg-Novi, Yugoslavia, Beograd 1969, 331-332.
Friedhelm Waldhausen, Eine Verallgemeinerung des Schleifensatzes, Topology 6 (1967), 501–504 (German). MR 220300, DOI 10.1016/0040-9383(67)90007-9

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Canonical splittings of groups and 3-manifolds
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Abstract: