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On the Rank of the Intersection of Subgroups of a Fuchsian Group

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Part of the Lecture Notes in Mathematics book series (LNM,volume 372)

Abstract

The Fuchsian groups are the discrete subgroups of LF(2, R), the group of all 2 × 2 matrices over the reels with determinant +1. We are interested here in the following group-theoretical property in particular in connection with Fuchsian groups: a group is said to have the finitely generated intersection property if the intersection of every pair of finitely generated subgroups is again finitely generated. In his paper [4], Greenberg proved, using geometrical methods, that Fuchsian groups have the finitely generated intersection property thereby extending the result of Howson [7] that free groups have the finitely generated intersection property. In [3] the present author, using purely algebraic methods, extended Greenberg’s result by giving a fairly general criterion for an amalgamated product of two groups to have the finitely generated intersection property. Here we show that the methods of [3] can he made to yield an explicit bound for the rank of the intersection of two subgroups of a Fuchsian group in terms of their ranks. (By the rank r(G) of a finitely generated group G we mean the smallest number of generators for G.)

Keywords

  • Cyclic Group
  • Initial Segment
  • Free Product
  • Fuchsian Group
  • Double Coset

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1974 Springer-Verlag Berlin Heidelberg

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Burns, R.G. (1974). On the Rank of the Intersection of Subgroups of a Fuchsian Group. In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21571-5_14

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  • DOI: https://doi.org/10.1007/978-3-662-21571-5_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06845-7

  • Online ISBN: 978-3-662-21571-5

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