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Residual properties of groups defined by basic commutators. (English) Zbl 1316.20032
Magnus 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.

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
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References:
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