Pro-Hall R-groups and groups discriminated by the free pro-p group
-
Montserrat Casals-Ruiz
,
Ilya Kazachkov
Abstract
In this note we introduce pro-Hall R-groups as inverse limits of Hall R-groups and show that for the binomial closure Sbin of any ring S discriminated by ℤp, the free pro-Hall Sbin-group 𝔽(A,Sbin) is fully residually free pro-p. Furthermore, we prove that any finite set of elements in 𝔽(A,Sbin) defines a pro-p subgroup and so an irreducible coordinate group over the free pro-p group.
Funding source: ERC
Award Identifier / Grant number: PCG-336983
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: N14-01-00068
Funding statement: The first author is supported by the Juan de la Cierva Programme of the Spanish Government. The second author is supported by the ERC grant PCG-336983. The first two authors are partly supported by the the Spanish Government, grant MTM2014-53810-C2-2-P, and by the Basque Government, grant IT974-16. The third author is supported by the grant of the Russian Fund for Basic Research N14-01-00068.
References
1 S. G. Afanaseva, Algebraic geometry over rigid solvable pro-p groups PhD thesis, Sobolev Mathematical Institute, 2014. Search in Google Scholar
2 N. Bourbaki, Commutative Algebra, Springer, Berlin, 1989. Search in Google Scholar
3 E. Daniyarova, A. Myasnikov and V. Remeslennikov, Unification theorems in algebraic geometry, Aspects of Infinite Groups, Algebra Discrete Math. 1, World Scientific, Hackensack (2008), 80–111. 10.1142/9789812793416_0007Search in Google Scholar
4 E. Daniyarova, A. Myasnikov and V. Remeslennikov, Algebraic geometry over algebraic systems. II: Foundations (in Russian), Fundam. Prikl. Mat. 17 (2011/12), 1, 65–106. 10.1134/S1064562411050073Search in Google Scholar
5 J. Elliott, Binomial rings, integer-valued polynomials, and λ-rings, J. Pure Applied Algebra 207 (2006), 1, 165–185. 10.1016/j.jpaa.2005.09.003Search in Google Scholar
6 P. Hall, Nilpotent Groups, Queen Mary College Math. Notes, Queen Mary College (University of London), London, 1969. Search in Google Scholar
7 O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over free group II: Systems in triangular quasi-quadratic form and description of residually free groups, J. Algebra 200 (1998), 2, 517–570. 10.1006/jabr.1997.7184Search in Google Scholar
8 D. H. Kochloukova and P. A. Zalesskii, On pro-p analogues of limit groups via extensions of centralizers, Math. Z. 267 (2011), 109–128. 10.1007/s00209-009-0611-ySearch in Google Scholar
9 R. C. Lyndon, Groups with parametric exponents, Trans. Amer. Math. Soc. 96 (1960), 518–533. 10.1090/S0002-9947-1960-0151502-6Search in Google Scholar
10 A. Malcev, Algebraic Structures (in Russian), Nauka, Moscow, 1976. Search in Google Scholar
11 S. G. Melesheva, Equations and algebraic geometry over profinite groups, Algebra Logic 49 (2010), 5, 444–455. 10.1007/s10469-010-9108-3Search in Google Scholar
12 R. B. Warfield, Nilpotent Groups, Lecture notes in Math. 513, Springer, Berlin, 1976. 10.1007/BFb0080152Search in Google Scholar
13 J. S. Wilson, Profinite Groups, Clarendon Press, Oxford, 1998. 10.1093/oso/9780198500827.001.0001Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
Articles in the same Issue
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
