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.

V3: A small correction in the last phrase of the proof of Theorem 1. 4 pages

References in corpus (3)

• How to generalize known results on equations over groups
• The Kervaire-Laudenbach conjecture and presentations of simple groups
• Diagrams and the second homotopy group

Cited by in corpus (6)

• Quasiperiodic and mixed commutator factorizations in free products of groups
• Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator
• The SQ-universality of one-relator relative presentations
• The structure of one-relator relative presentations and their centres
• Commutator length of powers in free products of groups
• Malnormality and centers in one-relator relative presentations