Intersecting free subgroups in free products of groups

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Abstract

A subgroup H of a free product Π*α∈I Gα of groups Gα, α ∈ I, is termed factor free if for every S ∈ 7Pi;*α∈I Gα and β ∈ I one has SHS-1 ∩ Gβ = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote r̄(K) = max(r(K) -1, 0), where r(K) is the rank of K. It has earlier been proved by the author that if H, K are finitely generated factor free subgroups of Π*α∈I Gα then r̄(H ∩ K) ≤ 6r̄(H)r̄(K). It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in Π*α∈I Gα so that r̄(H), r̄(K) > 0 and r̄(H ∩ K) = 6r̄(H)r̄(K). It is also proved that if the factors Gα, α ∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of Π*α∈I Gα then r̄(H ∩ K) ≤ 2r̄(H)r̄(K).

Original languageEnglish (US)
Pages (from-to)281-290
Number of pages10
JournalInternational Journal of Algebra and Computation
Volume11
Issue number3
DOIs
StatePublished - 2001

ASJC Scopus subject areas

  • General Mathematics

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