TY - JOUR

T1 - Intersecting free subgroups in free products of groups

AU - Ivanov, S. V.

N1 - Funding Information:
∗Supported in part by an Alfred P. Sloan Research Fellowship and NSF grant DMS 98-01500.

PY - 2001

Y1 - 2001

N2 - 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).

AB - 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).

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U2 - 10.1142/S0218196701000267

DO - 10.1142/S0218196701000267

M3 - Article

AN - SCOPUS:0035640926

SN - 0218-1967

VL - 11

SP - 281

EP - 290

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 3

ER -