On the verbal width of finitely generated pro-$p$ groups

Andrei Jaikin-Zapirain

Source: Rev. Mat. Iberoamericana Volume 24, Number 2 (2008), 617-630.

Abstract

Let $p$ be a prime. It is proved that a non-trivial word $w$ from a free group $F$ has finite width in every finitely generated pro-$p$ group if and only if $w\not \in (F^\prime)^{p} F^{\prime\prime}$. Also it is shown that any word $w$ has finite width in a compact $p$-adic group.

Primary Subjects: 20E18

Secondary Subjects: 22E35

Keywords: pro-$p$ group; verbal subgroup; verbal width; $p$-adic analytic group

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Permanent link to this document: http://projecteuclid.org/euclid.rmi/1218475356

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References

Barnea, Y. and Shalev, A.: Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras. Trans. Amer. Math. Soc. 349 (1997), 5073-5091.

Mathematical Reviews (MathSciNet): MR1422889

Digital Object Identifier: doi:10.1090/S0002-9947-97-01918-1

JSTOR: links.jstor.org

Bourbaki, N.: Lie groups and Lie algebras. Chapters 1-3. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1989.

Mathematical Reviews (MathSciNet): MR979493

Burns, R. G. and Medvedev, Yu.: Analytic relatively free pro-$p$ groups. J. Group Theory 7 (2004), 533-541.

Mathematical Reviews (MathSciNet): MR2080450

Digital Object Identifier: doi:10.1515/jgth.2004.7.4.533

Dixon, J., du Sautoy, M., Mann, A. and Segal, D.: Analytic pro-$p$ groups. Second edition. Cambridge Studies in Advanced Mathematics 61. Cambridge University Press, Cambridge, 1999.

Mathematical Reviews (MathSciNet): MR1720368

Govorov, V. E.: Graded algebras. Mat. Zametki 12 (1972), 197-204 (Russian). Translation in Math. Notes 12 (1972), 552-556.

Mathematical Reviews (MathSciNet): MR318199

Hartley, B.: Subgroups of finite index in profinite groups. Math. Z. 168 (1979), 71-76.

Mathematical Reviews (MathSciNet): MR542184

Digital Object Identifier: doi:10.1007/BF01214436

Huppert, B. and Blackburn, N.: Finite groups II. Fundamental Principles of Mathematical Sciences 242. Springer-Verlag, Berlin-New York, 1982

Mathematical Reviews (MathSciNet): MR650245

Ivanov, S. V.: P. Hall's conjecture on the finiteness of verbal subgroups. Izv. Vyssh. Uchebn. Zaved. Mat. 325 (1989), 60-70. Translation in Soviet Math. (Iz. VUZ) 33 (1989), 59-70.

Mathematical Reviews (MathSciNet): MR1017779

Lubotzky, A. and Segal, D.: Subgroup growth. Progress in Mathematics 212. Birkhäuser Verlag, Basel, 2003.

Mathematical Reviews (MathSciNet): MR1978431

Merzljakov, Ju. I.: Verbal and marginal subgroups of linear groups. (Russian) Dokl. Akad. Nauk SSSR 177 (1967) 1008-1011.

Mathematical Reviews (MathSciNet): MR222150

Neumann, H.: Varieties of groups. Springer-Verlag, New York, 1967.

Mathematical Reviews (MathSciNet): MR215899

Newman, M. F., Schneider, C. and Shalev, A.: The entropy of graded algebras. J. Algebra 223 (2000), 85-100.

Mathematical Reviews (MathSciNet): MR1738253

Digital Object Identifier: doi:10.1006/jabr.1999.8053

Nikolov, N. and Segal, D.: On finitely generated profinite groups. I: strong completeness and uniform bounds. II: products in quasisimple groups. Ann. of Math. (2) 165 (2007), 171-238, 239-273.

Mathematical Reviews (MathSciNet): MR2276769

Piontkovskiĭ, D. I.: Hilbert series and relations in algebras. Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 205-219 (Russian). Translation in Izv. Math. 64 (2000), 1297-1311.

Mathematical Reviews (MathSciNet): MR1817254

Roman'kov, V. A.: The width of verbal subgroups of solvable groups. Algebra i Logika 21 (1982), 60-72 (Russian). Translation in Algebra and Logic 21 (1982), 41-49.

Mathematical Reviews (MathSciNet): MR683939

Segal, D.: Closed subgroups of profinite groups. Proc. London Math. Soc. (3) 81 (2000), 29-54.

Mathematical Reviews (MathSciNet): MR1756331

Digital Object Identifier: doi:10.1112/S002461150001234X

Segal, D.: Variations on a theme of Burns and Medvedev. Groups Geom. Dyn. 1 (2007), no. 4, 661-668.

Mathematical Reviews (MathSciNet): MR2357487

Serre, J. P.: Lie algebras and Lie groups. Second edition. Lecture Notes in Mathematics 1500. Springer-Verlag, 1992.

Mathematical Reviews (MathSciNet): MR1176100

Shalev, A.: Characterization of $p$-adic analytic groups in terms of wreath products. J. Algebra 145 (1992), 204-208.

Mathematical Reviews (MathSciNet): MR1144667

Digital Object Identifier: doi:10.1016/0021-8693(92)90185-O

Stroud, P. W.: Topics in theory of verbal subgroups. PhD Thesis, University of Cambridge, 1966.

Wilson, J. S.: Two-generator conditions for residually finite groups. Bull. London Math. Soc. 23 (1991), 239-248.

Mathematical Reviews (MathSciNet): MR1123332

Digital Object Identifier: doi:10.1112/blms/23.3.239

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