Splittings of groups and intersection numbers

Scott, Peter¹ ; Swarup, Gadde A²

¹ Mathematics Department, University of Michigan, Ann Arbor, Michigan 48109, USA
² Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia

Geometry & topology, Tome 4 (2000) no. 1, pp. 179-218.

Résumé

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.

Détail

DOI : 10.2140/gt.2000.4.179

Keywords: amalgamated free product, splitting, intersection number, ends

Bibliographie

[1] G M Bergman, On groups acting on locally finite graphs, Ann. of Math. $(2)$ 88 (1968) 335

[2] D E Cohen, Groups of cohomological dimension one, Lecture Notes in Mathematics 245, Springer (1972)

[3] M J Dunwoody, Accessibility and groups of cohomological dimension one, Proc. London Math. Soc. $(3)$ 38 (1979) 193

[4] M J Dunwoody, M A Roller, Splitting groups over polycyclic-by-finite subgroups, Bull. London Math. Soc. 25 (1993) 29

[5] M J Dunwoody, E L Swenson, The algebraic torus theorem, Invent. Math. 140 (2000) 605

[6] M Freedman, J Hass, P Scott, Closed geodesics on surfaces, Bull. London Math. Soc. 14 (1982) 385

[7] M Freedman, J Hass, P Scott, Least area incompressible surfaces in 3–manifolds, Invent. Math. 71 (1983) 609

[8] P H Kropholler, M A Roller, Splittings of Poincaré duality groups, Math. Z. 197 (1988) 421

[9] P H Kropholler, M A Roller, Relative ends and duality groups, J. Pure Appl. Algebra 61 (1989) 197

[10] B Leeb, P Scott, A geometric characteristic splitting in all dimensions, Comment. Math. Helv. 75 (2000) 201

[11] W D Neumann, G A Swarup, Canonical decompositions of 3–manifolds, Geom. Topol. 1 (1997) 21

[12] M Sageev, Codimension–1 subgroups and splittings of groups, J. Algebra 189 (1997) 377

[13] P Scott, A new proof of the annulus and torus theorems, Amer. J. Math. 102 (1980) 241

[14] P Scott, The symmetry of intersection numbers in group theory, Geom. Topol. 2 (1998) 11

[15] P Scott, T Wall, Topological methods in group theory, from: "Homological group theory (Proc. Sympos., Durham, 1977)", London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137

[16] G P Scott, G A Swarup, An algebraic annulus theorem, Pacific J. Math. 196 (2000) 461

[17] P Scott, G A Swarup, Canonical splittings of groups and 3–manifolds, Trans. Amer. Math. Soc. 353 (2001) 4973

[18] Z Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, Geom. Funct. Anal. 7 (1997) 561

[19] J P Serre, Arbres, amalgames, $\mathrm{SL}_2$, Astérisque 46, Société Mathématique de France (1977)

[20] J P Serre, Trees, Springer (1980)

[21] J Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs 4, Yale University Press (1971)