## Abstract

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if H_{1} is a finitely generated factor-free noncyclic subgroup of the free product G_{1} G_{2} of two finite groups G_{1}, G_{2}, then the WN-coefficientσH_{1}/ of H_{1} is rational and can be computed in exponential time in the size of H_{1}. This coefficientσH_{1}/ is the minimal positive real number such that, for every finitely generated factor-free subgroup H_{2} of G_{1} G_{2}, it is true that Nr.H_{1};H_{2}/σH_{1}/Nr.H_{1}/Nr.H_{2}/, where Nr.H/ D max.r.H/ ≤ 1; 0/ is the reduced rank of H, r.H/ is the rank of H, and Nr.H_{1};H_{2}/ is the reduced rank of the generalized intersection of H_{1} and H_{2}. In the case of the free product G_{1} G_{2} of two finite groups G_{1}, G_{2}, it is also proved that there exists a factor-free subgroup H 2 D H 2σH_{1}/ such that Nr.H_{1};H 2 / DσH_{1}/Nr.H_{1}/Nr.H 2 /, H 2 has at most doubly exponential size in the size of H_{1}, and H 2 can be constructed in exponential time in the size of H_{1}.

Original language | English (US) |
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Pages (from-to) | 1113-1177 |

Number of pages | 65 |

Journal | Groups, Geometry, and Dynamics |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - 2017 |

## Keywords

- Free and factor-free subgroups
- Free products of groups
- Linear programming
- Rank of intersection of factor-free subgroups

## ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics