On the Kurosh rank of the intersection of subgroups in free products of groups

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Abstract

The Kurosh rank rK (H) of a subgroup H of a free product ∏α ∈ I* Gα of groups Gα, α ∈ I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H1, H2 are subgroups of ∏α ∈ I* Gα and H1, H2 have finite Kurosh rank, then over(r, ̄)K (H1 ∩ H2) ≤ 2 frac(q*, q* - 2) over(r, ̄)K (H1) over(r, ̄)K (H2) ≤ 6 over(r, ̄)K (H1) over(r, ̄)K (H2), where over(r, ̄)K (H) = max (rK (H) - 1, 0), q* is the minimum of orders >2 of finite subgroups of groups Gα, α ∈ I, q* : = ∞ if there are no such subgroups, and frac(q*, q* - 2) : = 1 if q* = ∞. In particular, if the factors Gα, α ∈ I, are torsion-free groups, then over(r, ̄)K (H1 ∩ H2) ≤ 2 over(r, ̄)K (H1) over(r, ̄)K (H2).

Original languageEnglish (US)
Pages (from-to)465-484
Number of pages20
JournalAdvances in Mathematics
Volume218
Issue number2
DOIs
StatePublished - Jun 1 2008

Keywords

  • Free products of groups
  • Kurosh rank of a subgroup of a free product
  • Subgroups of free products

ASJC Scopus subject areas

  • Mathematics(all)

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