Geometry & Topology

The symmetry of intersection numbers in group theory

Peter Scott

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For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

Article information

Geom. Topol., Volume 2, Number 1 (1998), 11-29.

Received: 21 February 1997
Revised: 13 March 1998
Accepted: 19 March 1998
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F32
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations 20E07: Subgroup theorems; subgroup growth 20E08: Groups acting on trees [See also 20F65] 57M07: Topological methods in group theory

ends amalgamated free products trees


Scott, Peter. The symmetry of intersection numbers in group theory. Geom. Topol. 2 (1998), no. 1, 11--29. doi:10.2140/gt.1998.2.11.

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  • D E Cohen, Groups of cohomological dimension one, Lecture Notes in Math. 245, Springer–Verlag, Berlin (1972)
  • M Dunwoody, M Sageev, JSJ splittings for finitely presented groups over slender groups, preprint
  • M H Freedman, J Hass, P Scott, Closed geodesics on surfaces, Bull. London Math. Soc. 14 (1982) 385–391
  • M H Freedman, J Hass, P Scott, Least area incompressible surfaces in 3–manifolds, Invent. Math. 71 (1983) 609–642.
  • C H Houghton, Ends of locally compact groups and their quotient spaces, J. Aust. Math. Soc. 17 (1974) 274–284
  • P H Kropholler, M A Roller, Splittings of Poincaré duality groups, Math. Zeit. 197 (1988) no. 3, 421–438
  • E Rips, Z Sela, Cyclic splittings of finitelypresented groups and the canonical JSJ decomposition, preprint
  • P Scott, Ends of pairs of groups, J. Pure Appl. Algebra 11 (1977) 179–198
  • P Scott, A new proof of the Annulus and Torus Theorems, Amer. J. Math. 102 (1980) 241–277
  • P Scott, G A Swarup, An Algebraic Annulus Theorem, preprint
  • P Scott, C T C Wall, Topological methods in group theory, from: “Homological Group Theory”, London Math. Soc. Lecture Notes Series 36 (1979) 137–214
  • J-P Serre, Arbres, amalgames, $SL_{2}$, Astérisque No. 46, Société Mathématique de France, Paris (1977)
  • J-P Serre, Trees, translated from French by John Stillwell, Springer–Verlag, Berlin–New York (1980) ISBN 3-540-10103-9