Geodesic
Linear programming and the intersection of free subgroups in free products of groups
Sergei V. Ivanov1
1University of Illinois-Urbana Champaign, USA
Groups, geometry, and dynamics, Tome 11 (2017) no. 4, pp. 1113-1177

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DOI

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 H1​ is a finitely generated factor-free noncyclic subgroup of the free product G1​∗G2​ of two finite groups G1​, G2​, then the WN-coefficient σ(H1​) of H1​ is rational and can be computed in exponential time in the size of H1​. This coefficient σ(H1​) is the minimal positive real number such that, for every finitely generated factor-free subgroup H2​ of G1​∗G2​, it is true that rˉ(H1​,H2​)≤σ(H1​)rˉ(H1​)rˉ(H2​), where rˉ(H) = max (r (H)−1,0) is the reduced rank of H, r(H) is the rank of H, and rˉ(H1​,H2​) is the reduced rank of the generalized intersection of H1​ and H2​. In the case of the free product G1​∗G2​ of two finite groups G1​, G2​, it is also proved that there exists a factor-free subgroup H2∗​=H2∗​(H1​) such that rˉ(H1​,H2∗​)=σ(H1​)rˉ(H1​)rˉ(H2∗​), H2∗​ has at most doubly exponential size in the size of H1​, and H2∗​ can be constructed in exponential time in the size of H1​.
DOI : 10.4171/ggd/424
Classification : 20-XX, 68-XX, 90-XX
Mots-clés : Free products of groups, free and factor-free subgroups, rank of intersection of factor-free subgroups, linear programming

Sergei V. Ivanov  1

1 University of Illinois-Urbana Champaign, USA 
Sergei V. Ivanov. Linear programming and the intersection of free subgroups in free products of groups. Groups, geometry, and dynamics, Tome 11 (2017) no. 4, pp. 1113-1177. doi: 10.4171/ggd/424
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     author = {Sergei V. Ivanov},
     title = {Linear programming and the intersection of free subgroups in free products of groups},
     journal = {Groups, geometry, and dynamics},
     pages = {1113--1177},
     year = {2017},
     volume = {11},
     number = {4},
     doi = {10.4171/ggd/424},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/424/}
}
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