TY - JOUR
T1 - On the Kurosh rank of the intersection of subgroups in free products of groups
AU - Ivanov, S. V.
N1 - Funding Information:
1 Supported in part by the NSF under grant DMS 04-00476.
PY - 2008/6/1
Y1 - 2008/6/1
N2 - 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).
AB - 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).
KW - Free products of groups
KW - Kurosh rank of a subgroup of a free product
KW - Subgroups of free products
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U2 - 10.1016/j.aim.2008.01.003
DO - 10.1016/j.aim.2008.01.003
M3 - Article
AN - SCOPUS:41049104151
SN - 0001-8708
VL - 218
SP - 465
EP - 484
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 2
ER -