Udc 512.543.7+512.543.16 Free Subgroups of One-relator Relative Presentations

Note that the existence of free subgroups in G̃ for n > 3 follows immediately from the free subgroup theorem for one-relator groups. Thus, Theorem 1 is nontrivial only for n = 2. The most difficult case is n = 1. An important role in this situation is played by the exponent sum of the generator in the relator. A word w = ∏ git εi ∈ G ∗ 〈t〉∞ is called unimodular if ∑ εi = 1. If the exponent sum of the generator in the word w equals to any number p 6= ±1, then it is unknown even whether the group G embeds into G̃ = 〈G, t | w = 1〉; in other words, it is unknown when the group G̃ is different from Z/pZ. There are a lot of papers on this subject, but the answer is known only under additional strong restrictions on the group G or/and on the word w (see, e.g., [B84], [KP95], [C02], [C03], [CG95], [CG00], [EH91], [FeR98], [GR62], [IK00], [Le62], [Ly80], [S87]). For this reason, in this paper we study only unimodular presentations. The injectivity of the natural mapping G → G̃ in the unimodular case was proved in [Kl93] (see also [FeR96]). More delicate properties of the group G̃ can be found in [CR01], [FoR03], [Kl06], and [Kl05].