Residual properties of groups defined by basic commutators.

*(English)*Zbl 1316.20032Magnus proved that free groups are residually torsion-free-nilpotent. This paper concerns residual torsion-free-nilpotence and other residual properties of families of (mainly one-relator) groups.

The first main result is that the Hydra groups \(G(k,a,t)=\langle a,t\mid [a,\underbrace{t,\ldots,t}_k]=1\rangle\) are residually torsion-free-nilpotent for all \(k\geq 1\). (A theorem of P. Hall [Ill. J. Math. 2, 787-801 (1958; Zbl 0084.25602)] is used in the proof.)

The following generalization of the Hydra groups is then considered. Let \(X,Y\) be disjoint sets of generators and let \(F\) be the free group on \(X\cup Y\). Let \(u,v\) be elements in the subgroups of \(F\) generated by \(X\) and \(Y\), respectively, and for each \(k\geq 1\) define \(G(k,u,v)=\langle X\cup Y\mid [u,\underbrace{v,\ldots,v}_k]=1\rangle\). Invoking a theorem of G. Kim and J. McCarron [J. Algebra 162, No. 1, 1-11 (1993; Zbl 0804.20024)], it is shown that if \(u,v\) are not proper powers then for each \(k>1\) the group \(G(k,u,v)\) is residually a finite \(p\)-group for every prime \(p\). Combining this with a theorem of J. P. Labute [J. Algebra 14, 16-23 (1970; Zbl 0198.34601)] it is shown that if \(u,v\) are basic commutators then for each \(k>1\) the group \(G(k,u,v)\) is residually torsion-free-nilpotent. Similar results (the second requiring additional hypotheses) are proved in the case \(k=1\). A further result (whose proof again requires the theorem of Labute) is that if \(u,v\) are basic commutators then the cyclically pinched one-relator group \(\langle x_1,\ldots,x_m,y_1,\ldots,y_n\mid u(x_1,\ldots,x_m)=v(y_1,\ldots,y_n)\rangle\) is residually torsion-free-nilpotent.

Finally, some quotients of the Hydra groups, defined by presentations with two generators and two relators (each of which is a basic commutator) are considered. Some of these are residually torsion-free-nilpotent, others are not. In particular, \(\langle a,t\mid [a,t,t]=[a,t,a,a,a]\rangle\) is not.

The first main result is that the Hydra groups \(G(k,a,t)=\langle a,t\mid [a,\underbrace{t,\ldots,t}_k]=1\rangle\) are residually torsion-free-nilpotent for all \(k\geq 1\). (A theorem of P. Hall [Ill. J. Math. 2, 787-801 (1958; Zbl 0084.25602)] is used in the proof.)

The following generalization of the Hydra groups is then considered. Let \(X,Y\) be disjoint sets of generators and let \(F\) be the free group on \(X\cup Y\). Let \(u,v\) be elements in the subgroups of \(F\) generated by \(X\) and \(Y\), respectively, and for each \(k\geq 1\) define \(G(k,u,v)=\langle X\cup Y\mid [u,\underbrace{v,\ldots,v}_k]=1\rangle\). Invoking a theorem of G. Kim and J. McCarron [J. Algebra 162, No. 1, 1-11 (1993; Zbl 0804.20024)], it is shown that if \(u,v\) are not proper powers then for each \(k>1\) the group \(G(k,u,v)\) is residually a finite \(p\)-group for every prime \(p\). Combining this with a theorem of J. P. Labute [J. Algebra 14, 16-23 (1970; Zbl 0198.34601)] it is shown that if \(u,v\) are basic commutators then for each \(k>1\) the group \(G(k,u,v)\) is residually torsion-free-nilpotent. Similar results (the second requiring additional hypotheses) are proved in the case \(k=1\). A further result (whose proof again requires the theorem of Labute) is that if \(u,v\) are basic commutators then the cyclically pinched one-relator group \(\langle x_1,\ldots,x_m,y_1,\ldots,y_n\mid u(x_1,\ldots,x_m)=v(y_1,\ldots,y_n)\rangle\) is residually torsion-free-nilpotent.

Finally, some quotients of the Hydra groups, defined by presentations with two generators and two relators (each of which is a basic commutator) are considered. Some of these are residually torsion-free-nilpotent, others are not. In particular, \(\langle a,t\mid [a,t,t]=[a,t,a,a,a]\rangle\) is not.

Reviewer: Gerald Williams (Colchester)

##### MSC:

20E26 | Residual properties and generalizations; residually finite groups |

20F05 | Generators, relations, and presentations of groups |

20F12 | Commutator calculus |

20F19 | Generalizations of solvable and nilpotent groups |

##### Keywords:

commutator calculus; residual nilpotence; basic commutators; one-relator groups; Hydra groups
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\textit{G. Baumslag} and \textit{R. Mikhailov}, Groups Geom. Dyn. 8, No. 3, 621--642 (2014; Zbl 1316.20032)

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