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A survey of recent results in groups and orderings: word problems, embeddings and amalgamations

Published online by Cambridge University Press:  05 July 2011

By
V. V. Bludov  and
A. M. W. Glass
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
V. V. Bludov
Affiliation:
East Siberia State Academy of Education, Russia
A. M. W. Glass
Affiliation:
Queens' College,Cambridge and Centre for Mathematical Sciences, Cambridge, England
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Introduction

This year we celebrate the centenaries of the births of B. H. Neumann and A. I. Mal'cev, two great pioneers in the study of ordered algebraic structures. This paper is a survey of our recent progress in this area, and amplifies the talks we gave in Bath at the Groups St. Andrews Conference there, and the first author gave at the Mal'cev Centenary Conference in Novosibirsk. A version of this paper in Russian will appear in honour of the Mal'cev Centenary and the 80th anniversary of A. I. Kokorin, another pioneer in the subject (Izvestia of Irkutsk Sate University, Series Matematika, Vol. 2 (2009), No. 2, 4–19). The heart of the paper for pure group theorists is Section 4.

Groups versus Lattice-ordered Groups

One way to obtain results in infinite group theory is through spelling and associated constructions, such as free groups, free products, free products with amalgamated subgroups, and HNN-extensions [23]. The first two constructions are available in any variety of algebras, but for groups they are especially nice. This is for two reasons. The first is that every element of a free group has a unique easily obtained reduced spelling in terms of the generators and there is, similarly, a unique normal form for elements of a free product. Moreover, given any groups G1 and G2 with isomorphic subgroups H1 and H2, respectively (say φ : H1 ≅ H2), there is a group L and embeddings Τi : Gi → L (i = 1, 2) such that h1Τ1 = h1φΤ2 for all h1 ∈ H1.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 150 - 160
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] S., Aanderaa, A proof of Higman's Embedding Theorem using Britton extensions of groups, in Word Problems, Decision Problems, and the Burnside Problem (ed. W. W., Booneet al), North Holland, Amsterdam, 1973, 1–18.Google Scholar
[2] G. M., Bergman, Ordering coproducts of groups and semigroups, J. Algebra 133 (1990), 313–339.Google Scholar
[3] V. V., Bludov, On the free product of right-ordered groups with amalgamated subgroups, Problems of modern mathematics, Proceedings of scientific works, V. II, NII MIOO NGU, Novosibirsk, 1966, 30–35 (in Russian).Google Scholar
[4] V. V., Bludov, M., Giraudet, A. M. W., Glass and G., Sabbagh, Automorphism groups of models of first order theories, in Models, Modules and Abelian Groups: In Memory of A. L. S. Corner (ed. R., Göbel and B., Goldsmith), W. de Gruyter, Berlin 2008, 329–332.Google Scholar
[5] V. V., Bludov and A. M. W., Glass, Conjugacy in lattice-ordered and right orderable groups, J. Group Theory 11 (2008), 623–633.Google Scholar
[6] V. V., Bludov and A. M. W., Glass, Free products of right ordered groups with amalgamated subgroups, Math. Proc. Cambridge Philos. Soc. 146 (2009), 591–601.Google Scholar
[7] V. V., Bludov and A. M. W., Glass, Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. 2009, doi: 10.1112/plms/pdp008.Google Scholar
[8] V. V., Bludov and A. M. W., Glass, Right orders and amalgamation for lattice-ordered groups, Math. Slovaca (to appear).
[9] W. W., Boone and G., Higman, An algebraic characterization of the solvability of the word problem, J. Austral. Math. Soc. 18 (1974), 41–53.Google Scholar
[10] J. L., Britton, The word problem, Ann. of Math. 77 (1963), 16–32.Google Scholar
[11] A. M. W., Glass, Partially Ordered Groups, Series in Algebra 7, World Scientific Pub. Co., Singapore, 1999.
[12] A. M. W., Glass, Sublattice subgroups of finitely presented lattice-ordered groups, J. Algebra 301 (2006), 509–530.Google Scholar
[13] A. M. W., Glass, Finitely generated lattice-ordered groups with soluble word problem, J. Group Theory 11 (2008), 1–21.Google Scholar
[14] A. M. W., Glass and Y., Gurevich, The word problem for lattice-ordered groups, Trans. Amer. Math. Soc. 280 (1983), 127–138.Google Scholar
[15] G., Higman, Subgroups of finitely presented groups, Proc. Royal Soc. London Ser. A 262 (1961), 455–475.Google Scholar
[16] W. C., Holland, The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408.Google Scholar
[17] W. C., Holland and E., Scrimger, Free products of lattice-ordered groups, Algebra Universalis 2 (1972), 247–254.Google Scholar
[18] A. I., Kokorin and V. M., Kopytov, Linearly Ordered Groups, Halstead Press, New York, 1974.Google Scholar
[19] V. M., Kopytov and N. Ya., Medvedev, Right-Ordered Groups, Plenum Pub. Co., New York, 1996.Google Scholar
[20] V. M., Kopytov and N. Ya., Medvedev, Selected Problems of Algebra, in Ordered groups: a Collection of works dedicated to the memory of N. Ya. Medvedev, Altai State University, Barnaul, 2007, 15–113 (in Russian).Google Scholar
[21] S., Lemieux, Conjugacy Problem: Open Questions and an Application, Ph.D. Thesis, University of Alberta, 2004, Edmonton, Alberta, Canada.
[22] P. A., Linnell, Left ordered groups with no non-abelian free subgroups, J. Group Theory 4 (2001), 153–168.Google Scholar
[23] R. C., Lyndon and P. E., Schupp, Combinatorial Group Theory, Ergeb. der Math. und ihr. Grenzgebiete 89, Springer-Verlag, Heidelberg, 1977.
[24] K. R., Pierce, Amalgamations of lattice-ordered groups, Trans. Amer. Math. Soc. 172 (1972), 249–260.Google Scholar
[25] A. A., Vinogradov, On the free product of ordered groups, Math. Sbornik 25 (1949), 163–168 (in Russian).Google Scholar

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