Geodesic
Free subgroups of one-relator relative presentations
Algebra i logika, Tome 46 (2007) no. 3, pp. 290-298.

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Suppose that $G$ is a non-trivial torsion-free group and $w$ is a word over the alphabet $G\cup\{x^{\pm1}_1,\dots,x^{\pm1}_n\}$. It is proved that, for $n\geqslant2$, the group $\widetilde G=\langle G,x_1,x_2,\dots,x_n\,|\,w = 1\rangle$ always contains a non-Abelian free subgroup. For $n=1$, the question whether there exist non-Abelian free subgroups in $\widetilde G$ is amply settled for the unimodular case (i.e., where the exponent sum of $x_1$ in $w$ is one). Some generalizations of these results are discussed.
Keywords: relative presentations, one-relator groups, free subgroups. 
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A. A. Klyachko. Free subgroups of one-relator relative presentations. Algebra i logika, Tome 46 (2007) no. 3, pp. 290-298. http://geodesic.mathdoc.fr/item/AL_2007_46_3_a1/

[1] W. Magnus, “Das Identitätsproblem für Gruppen mit einer definierenden Relation”, Math. Ann., 106 (1932), 295–307 | DOI | MR | Zbl

[2] D. I. Moldavanskii, “Ob odnoi teoreme Magnusa”, Uch. zap. Ivanovsk. gos. ped. in-ta, 44 (1969), 26–28 | MR

[3] S. D. Brodskii, “Uravneniya nad gruppami i gruppy s odnim opredelyayuschim sootnosheniem”, Sib. matem. zh., 25:2 (1984), 84–103 | MR | Zbl

[4] A. A. Klyachko, M. I. Prischepov, “Metod spuska dlya uravnenii nad gruppami”, Vestn. MGU, matem., mekh., 1995, no. 4, 90–93 | MR | Zbl

[5] A. Clifford, “A class of exponent sum two equations over groups”, Glasg. Math. J., 44:2 (2002), 201–207 | DOI | MR | Zbl

[6] A. Clifford, “Nonamenable type K equations over groups”, Glasg. Math. J., 45:2 (2003), 389–400 | DOI | MR | Zbl

[7] A. Clifford, R. Z. Goldstein, “Tesselations of $S^2$ and equations over torsion-free groups”, Proc. Edinb. Math. Soc., II Ser., 38:3 (1995), 485–493 | DOI | MR | Zbl

[8] A. Clifford, R. Z. Goldstein, “Equations with torsion-free coefficients”, Proc. Edinb. Math. Soc., II Ser., 43:2 (2000), 295–307 | DOI | MR | Zbl

[9] M. Edjvet, J. Howie, “The solution of length four equations over groups”, Trans. Am. Math. Soc., 326:1 (1991), 345–369 | DOI | MR | Zbl

[10] R. Fenn, C. Rourke, “Characterisation of a class of equations with solution over torsion-free groups”, The Epstein Birthday Schrift, ded. to D. Epstein on the occasion of his 60th birthday, Geom. Topol. Monogr., 1, eds. I. Rivin et al., Univ. Warwick, Inst. Math., Warwick, 1998, 159–166 | MR | Zbl

[11] M. Gerstenhaber, O. S. Rothaus, “The solution of sets of equations in groups”, Proc. Natl. Acad. Sci. USA, 48 (1962), 1531–1533 | DOI | MR | Zbl

[12] S. V. Ivanov, A. A. Klyachko, “Solving equations of length at most six over torsion-free groups”, J. Group Theory, 3:3 (2000), 329–337 | DOI | MR | Zbl

[13] F. Levin, “Solutions of equations over groups”, Bull. Am. Math. Soc., 68 (1962), 603–604 | DOI | MR | Zbl

[14] R. C. Lyndon, “Equations in groups”, Bol. Soc. Bras. Math., 11:1 (1980), 79–102 | DOI | MR | Zbl

[15] J. R. Stallings, “A graph-theoretic lemma and group embeddings”, Combinatorial group theory and topology, Sel. pap. conf., Alta/Utah 1984, Ann. Math. Stud., 111, 1987, 145–155 | MR | Zbl

[16] A. A. Klyachko, “A funny property of a sphere and equations over groups”, Comm. Algebra, 21:7 (1993), 2555–2575 | DOI | MR | Zbl

[17] R. Fenn, C. Rourke, “Klyachko's methods and the solution of equations over torsion-free groups”, Enseign. Math., II Ser., 42:1–2 (1996), 49–74 | MR | Zbl

[18] V. V. Cohen, C. Rourke, “The surjectivity problem for one-generator, one-relator extensions of torsion-free groups”, Geom. Topol., 5 (2001), 127–142 | DOI | MR | Zbl

[19] M. Forester, C. Rourke, Diagrams and the second homotopy group, arXiv: math.AT/0306088 | MR

[20] A. A. Klyachko, “Kak obobschit izvestnye rezultaty ob uravneniyakh nad gruppami”, Matem. zam., 79:3 (2006), 409–419 ; arXiv: math.GR/0406382 | MR | Zbl

[21] A. A. Klyachko, “Gipoteza Kervera-Laudenbakha i kopredstavleniya prostykh grupp”, Algebra i logika, 44:4 (2005), 399–437 | MR | Zbl

[22] S. D. Promyslow, “A simple example of a torsion free nonunique product group”, Bull. Lond. Math. Soc., 20 (1988), 302–304 | DOI | MR

[23] E. Rips, Y. Segev, “Torsion free groups without unique product property”, J. Algebra, 108 (1987), 116–126 | DOI | MR | Zbl

[24] P. Lindon, P. Shupp, Kombinatornaya teoriya grupp, Mir, M., 1980 | MR