Torsion in profinite completions. (English) Zbl 0829.20044
Let \(G\) be a residually finite torsion-free group and let \(\widehat{G}\) be the profinite completion of \(G\). It was shown by M. J. Evans [J. Pure Appl. Algebra 65, 101-104 (1990; Zbl 0705.20020)] that \(\widehat{G}\) need not be a torsion-free group, and subsequently A. Lubotzky [Q. J. Math., Oxf. II. Ser. 44, 327-332 (1993; see the preceding review Zbl 0829.20043)] showed that \(\widehat{G}\) can contain a copy of every finite group.
The authors study torsion in \(\widehat{G}\) when \(G\) is soluble. They show that \(\widehat{G}\) is torsion free if \(G\) is an abelian group, a finitely generated abelian by nilpotent or soluble minimax group, but that \(\widehat{G}\) can have torsion if \(\widehat{G}\) is a nilpotent group of class 2. Their main result, proved by a clever and striking construction, is that \(\widehat{G}\) can have torsion if \(G\) is a finitely generated centre by metabelian group.
The authors study torsion in \(\widehat{G}\) when \(G\) is soluble. They show that \(\widehat{G}\) is torsion free if \(G\) is an abelian group, a finitely generated abelian by nilpotent or soluble minimax group, but that \(\widehat{G}\) can have torsion if \(\widehat{G}\) is a nilpotent group of class 2. Their main result, proved by a clever and striking construction, is that \(\widehat{G}\) can have torsion if \(G\) is a finitely generated centre by metabelian group.
Reviewer: P.Zalesskij (Minsk)
MSC:
| 20E18 | Limits, profinite groups |
| 20F16 | Solvable groups, supersolvable groups |
| 20E26 | Residual properties and generalizations; residually finite groups |
| 20E07 | Subgroup theorems; subgroup growth |
Keywords:
residually finite torsion-free groups; profinite completions; Abelian groups; soluble minimax groups; nilpotent groups; finitely generated centre by metabelian groupsReferences:
| [1] | Crawley-Boevey, W. W.; Kropholler, P. H.; Linnell, P. A., Torsion-free soluble groups, completions and the zero divisor conjecture, J. Pure Appl. Algebra, 54, 181-196 (1988) · Zbl 0666.16007 |
| [2] | Evans, M. J., Torsion in pro-finite completions of torsion-free groups, J. Pure Appl. Algebra, 65, 101-104 (1990) · Zbl 0705.20020 |
| [3] | Hall, P., On the finiteness of certain soluble groups, Proc. London Math. Soc., 9, 3, 595-622 (1959) · Zbl 0091.02501 |
| [4] | A. Lubotsky, Torsion in profinite completions, Quart. J. Math. Oxford, to appear.; A. Lubotsky, Torsion in profinite completions, Quart. J. Math. Oxford, to appear. |
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