2021/01/30 by Feldkamp, Carsten
#20E06 #20E07 #20F05 #20F34 #20F70 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2102.00285
In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let F be a free group with basis X and let r be a cyclically reduced element of F which contains a basis element x ∈ X, then every non-trivial element of the normal closure of r in F contains the basis element x. Equivalently, the subgroup freely generated by X \backslash \x\ embeds canonically into the quotient group F / ⟨ ⟨ r ⟩ ⟩F. In this article, we want to introduce a Freiheitssatz for amalgamated products G=A ∗U B of free groups A and B, where U is a maximal cyclic subgroup in A and B: If an element r of G is neither conjugate to an element of A nor B, then the factors A, B embed canonically into G / ⟨ ⟨ r ⟩ ⟩G.