zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] G. Baumslag, On generalised free products. Math. Z. 78 (1962), 423-438. · Zbl 0104.24402
[2] G. Baumslag, Groups with the same lower central sequence as a relatively free group. I. Trans. Amer. Math. Soc. 129 (1967), 308-321. · Zbl 0153.35002
[3] G. Baumslag, On the residual nilpotence of certain one-relator groups. Comm. Pure Appl. Math. 21 (1968), 491-506. · Zbl 0186.32101
[4] G. Baumslag, Recognizing powers in nilpotent groups and nilpotent images of free groups. J. Aust. Math. Soc. 83 (2007), no. 2, 149-155. · Zbl 1146.20030
[5] G. Baumslag, Lecture notes on nilpotent groups. Regional Conference Series in Mathematics. American Mathematical Society, Providence, 1971. · Zbl 0241.20001
[6] G. Baumslag, Finitely generated residually torsion-free nilpotent groups. I. J. Austral. Math. Soc. Ser. A 67 (1999), no. 3, 289-317. · Zbl 0944.20021
[7] G. Baumslag, Recognizing powers in nilpotent groups and nilpotent images of free groups. J. Aust. Math. Soc. 83 (2007), no. 2, 149-155. · Zbl 1146.20030
[8] G. Baumslag, Some reflections on proving groups residually torsion-free nilpotent. I. Illinois J. Math. 54 (2010), 315-325. · Zbl 1250.20021
[9] G. Baumslag and R. Mikhailov, On residual properties of generalized Hydra groups. Preprint 2013. · Zbl 1316.20032
[10] G. Baumslag and H. Short, Torsion free hyperbolic groups are big powers groups. Unpublished.
[11] A. K. Bousfield, Homological localization towers for groups and \cdots -modules. Mem. Amer. Math. Soc. 10 (1977), no. 186, vii+68 pages. · Zbl 0364.20058
[12] T. Cochran and K. Orr, Stability of lower central series of compact 3-manifold groups. Topology 37 (1998), no. 3, 497-526. · Zbl 0902.57012
[13] W. Dison and T. Riley, Hydra groups. Comment. Math. Helv. 88 (2013), no. 3, 507-540. · Zbl 1305.20052
[14] D. Jackson, A. Gaglione, and D. Spellman, Basic commutators as relators. J. Group theory 5 (2002), no. 3, 351-363. · Zbl 1001.20028
[15] K. W. Gruenberg, Residual properties of infinite solvable groups. Proc. London Math. Soc. (3) 7 (1957), 29-62. · Zbl 0077.02901
[16] P. Hall, A contribution to the theory of groups of prime-power order. Proc. London Math. Soc. (2) 36 (1934), no. 1, 29-95. · Zbl 0007.29102
[17] P. Hall, Some sufficient conditions for a group to be nilpotent. Illinois J. Math. 2 (1958), 787-801. · Zbl 0084.25602
[18] M. Hall, Jr., The theory of groups. Second ed. Chelsea Publishing Company, New York, N.Y., 1976. · Zbl 0354.20001
[19] B. Hartley, Proc. London Math. Soc. (3) 20 (1970), 365-392. · Zbl 0194.03402
[20] G. Higman, Amalgams of p-groups. J. Algebra 1 (1964), 301-305. · Zbl 0246.20015
[21] P. Hilton, Nilpotent actions on nilpotent groups. In J. N. Crossley, Algebra and logic. Papers from the XIV Summer Research Institute of the Australian Mathematical Society; Monash University, Clayton, 1974. Springer, Berlin etc., 1975, 174-196. · Zbl 0305.20016
[22] G. Kim and J. McCarron, On amalgamated free-products of residually p-finite groups. J. Algebra 162 (1993), no. 1, 1-11. · Zbl 0804.20024
[23] J. P. Labute, On the descending central series of groups with a single defining relation. J. Algebra 14 (1970), 16-23. · Zbl 0198.34601
[24] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. École Norm. Sup. (3) 71 (1954), 101-190. · Zbl 0055.25103
[25] F. W. Levi, Über die Untergruppen freier Gruppen. Math. Z. 32 (1930), 315-318. · JFM 56.0134.01
[26] A. I. Lichtman, The residual nilpotence of the multiplicative group of a skew field generated by universal enveloping algebras. J. Algebra 112 (1988), no. 1, 250-263. · Zbl 0636.16007
[27] R. C. Lyndon, Groups with parametric exponents. Trans. Amer. Math. Soc. 96 (1960), 518-533. · Zbl 0108.02501
[28] W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring. Mat. Ann. 111 (1935), no. 1, 259-280. · Zbl 0011.15201
[29] W. Magnus, Über Gruppen und Zugeordnete Liesche Ringe. J. Reine Angew. Math. 182 (1940), 142-149. · Zbl 0025.24201
[30] A. Karrass, W. Magnus, and D. Solitar, Elements of finite order in groups with a single defining relation. Comm. Pure Appl. Math. 13 (1960), 57-66. · Zbl 0091.02403
[31] A. Karrass, W. Magnus, and D. Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations. Reprint of the 1976 second edition. Dover Publications, Inc., Mineola, N.Y., 2004. · Zbl 1130.20307
[32] A. Mal’cev, Generalized nilpotent groups and their adjoint groups. Mat. Sbornik N.S. 25(67) (1949), 347-366. In Russian.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.