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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 is a finitely generated factor-free noncyclic subgroup of the free product of two finite groups , , then the WN-coefficient of is rational and can be computed in exponential time in the size of . This coefficient is the minimal positive real number such that, for every finitely generated factor-free subgroup of , it is true that , where = max (r is the reduced rank of , r is the rank of , and is the reduced rank of the generalized intersection of and . In the case of the free product of two finite groups , , it is also proved that there exists a factor-free subgroup such that , has at most doubly exponential size in the size of , and can be constructed in exponential time in the size of .
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Sergei V. Ivanov, Linear programming and the intersection of free subgroups in free products of groups. Groups Geom. Dyn. 11 (2017), no. 4, pp. 1113–1177