Free subgroups of one-relator relative presentations

Abstract

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G=<G,x_1,x_2,...,x_n | w=1> always contains a nonabelian free subgroup. For n=1 the question about the existence of nonabelian free subgroups in \~G is answered completely in the unimodular case (i.e., when the exponent sum of x_1 in w is one). Some generalisations of these results are discussed.Comment: V3: A small correction in the last phrase of the proof of Theorem 1. 4 page

The structure of one-relator relative presentations and their centres

“…The same situation was considered in [15], [16] and [17] (and also in [6], [8], [9], [13] and [14] in the case n ¼ 1).…”

“…the Tits alternative holds for e G G (i.e., e G G either contains a non-abelian free subgroup or is virtually solvable) provided that it holds for G; see [17].…”

The relative hyperbolicity of one-relator relative presentations

“…In [7] it was proved thatĜ G contains a non-abelian free subgroup (and hence is non-elementary), except in the following two cases:…”

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

“…-G embeds (naturally) into G [Kl93] (see also [FeR96]);* ) -G is torsion-free [FoR05]; -G is not simple if it does not coincide with G [Kl05]; -G almost always (with some known exceptions) contains a non-abelian free subgroup [Kl07]; -G is SQ-universal if G decomposes nontrivially into a free product [Kl06b]; -the centre of G is almost always (with some known exceptions) trivial [Kl09]. Some generalisations of these results to relative presentations with several additional generators can be found in [Kl09], [Kl07], [Kl06a], and [Kl06b].…”