### Abstract

Let (G
_{ℓ}
ℓL) be a family of groups and let F be a free group. Let G denote F
_{ℓL}
G
_{ℓ}
, the free product of F and all the G
_{ℓ}
. Let denote the set of all finitely generated (free) subgroups H of G which have the property that, for each or each free group H, the reduced rank of H is defined as Set D is a finite subgroup of G with, for all We are interested in precise bounds for σ. If every element of is cyclic, then σ=0. We henceforth assume that some element of has rank two. In the case where G=F and, hence, θ=1, Hanna Neumann and Walter Neumann proved that σ[1,2] and it is a famous conjecture that σ=1, called the Strengthened Hanna Neumann Conjecture. For the general case, we proved that σ[1,2] and if G has 2-torsion then σ=2. We conjectured that if G is 2-torsion-free then σ=1. In this article, we prove the following implications which show that under certain circumstances σ<2. If G is 2-torsion-free and has 3-torsion, then σ≤8/7. If G is 2-torsion-free and 3-torsion-free and has 5-torsion, then σ≤9/5. If p is an odd prime number and G=C
_{p}
*C
_{p}
, then. In particular, if G=C
_{3}
*C
_{3}
then σ=1, and if G=C
_{5}
*C
_{5}
then σ≤1.52.

Original language | English (US) |
---|---|

Pages (from-to) | 223-248 |

Number of pages | 26 |

Journal | Illinois Journal of Mathematics |

Volume | 54 |

Issue number | 1 |

State | Published - Dec 1 2010 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Illinois Journal of Mathematics*,

*54*(1), 223-248.