Magnus Intersections in One-Relator Products

摘要

Recall that a group G is locally indicable if each of its nontrivial, finitely generated subgroups admits an infinite cyclic homomorphic image. A one-relator product of groups A_λ (λ ∈ Λ) is a quotient G = (*_(λ ∈ Λ)A_λ)/< < R > > of their free product by the normal closure of a word R, which is called the relator, and is assumed not to be conjugate to an element of one of the A_λ. Let A_Λ := *_(λ ∈ Λ)A_λ. Then, by the Freiheitssatz for locally indicable groups (see Theorem 2.1 to follow), a free factor A_M := *_(μ∈M)A_ μ of A_Λ embeds in G provided that R is not conjugate in A_Λ to an element of A_M . The image of this embedding is called the Magnus subgroup corresponding to the subset M is contained in Λ .