Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-788cddb947-tr9hg Total loading time: 0 Render date: 2024-10-08T08:38:21.090Z Has data issue: false hasContentIssue false

Discriminating groups: a comprehensive overview

Published online by Cambridge University Press:  05 July 2011

By
Benjamin Fine ,
Anthony M. Gaglione ,
Alexei Myasnikov ,
Gerhard Rosenberger  and
Dennis Spellman
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Benjamin Fine
Affiliation:
Fairfield University
Anthony M. Gaglione
Affiliation:
U.S. Naval Academy
Alexei Myasnikov
Affiliation:
McGill University
Gerhard Rosenberger
Affiliation:
University of Dortmund
Dennis Spellman
Affiliation:
Fairfield University
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
Get access

Summary

Abstract

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by Kharlampovich and Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

Introduction

Discriminating groups were introduced by Baumslag, Myasnikov and Remeslennikov as an outgrowth of their theory of algebraic geometry over groups. Algebraic geometry over groups was the main method of attack used by O. Kharlampovich and A. Myasnikov in their solution of the celebrated Tarski conjectures. The class of discriminating groups, however, has taken on a life of its own and has been an object of a considerable amount of study. In this paper we survey the large array of results concerning the class of discriminating groups that have been developed over the past decade.

In Section 1, we define discrimination for groups and describe its ties to other areas. Also the concept of trivially discriminating (TD) groups is introduced, and the concept of squarelike groups is defined. It is also indicated how to define discrimination for arbitrary algebraic systems. It is also shown how to generalize the concept of squarelike to arbitrary algebras.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 395 - 414
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] G., Baumslag, Lecture notes on nilpotent groups (Amer. Math. Soc., Providence, RI 1969).Google Scholar
[2] G., Baumslag, B., Fine, A. M., Gaglione and D., Spellman, Reflections on discriminating groups, J. Group Theory 10 (2007), 87–99.Google Scholar
[3] G., Baumslag, B., Fine, A. M., Gaglione and D., Spellman, A note on nondiscrimination of nilpotent groups and Malcev completions, Contemp. Math. 421 (2006), 29–34.Google Scholar
[4] G., Baumslag and C. F., Miller III, Reflections on some groups of B. H. Neumann, preprint.
[5] G., Baumslag, A. G., Myasnikov and V. N., Remeslennikov, Algebraic geometry over groups 1, J. Algebra 219 (1999), 16–79.Google Scholar
[6] G., Baumslag, A. G., Myasnikov and V. N., Remeslennikov, Discriminating and codiscriminating groups, J. Group Theory 3 (2000), 467–479.Google Scholar
[7] O., Belegradek, Discriminating and square-like groups, J. Group Theory 7 (2004), 521–532.Google Scholar
[8] O., Belegradek, The theory of square-like abelian groups is decidable, in Algebra, logic, set theory, Stud. Log. (Lond.) 4 (Coll. Publ., London 2007), 33–46.Google Scholar
[9] J. L., Bell and A. B., Slomson, Models and ultraproducts: an introduction (Second revised printing) (North-Holland, Amsterdam 1972).Google Scholar
[10] P. M., Cohn, Universal algebra (Harper and Row, New York, NY 1965).Google Scholar
[11] V. H., Dyson, private communication.
[12] B., Fine, A. M., Gaglione, A. G., Myasnikov and D., Spellman, Discriminating groups, J. Group Theory 4 (2001), 463–474.Google Scholar
[13] B., Fine, A. M., Gaglione, A. G., Myasnikov and D., Spellman, Groups whose universal theory is axiomatizable by quasi-identities, J. Group Theory 5 (2002), 365–381.Google Scholar
[14] B., Fine, A. M., Gaglione and D., Spellman, Every abelian group universally equivalent to a discriminating group is elementarily equivalent to a discriminating group, Contemp. Math. 296 (2002), 129–137.Google Scholar
[15] B., Fine, A. M., Gaglione and D., Spellman, The axiomatic closure of the class of discriminating groups, Arch. Math. (Basel) 83 (2004), 106–112.Google Scholar
[16] B., Fine, A. M., Gaglione and D., Spellman, Discriminating and squarelike groups I: axiomatics, Contemp. Math 360 (2004), 35–46.Google Scholar
[17] B., Fine, A. M., Gaglione and D., Spellman, Discriminating and squarelike groups II: examples, Houston J. Math. 31 (2005), 649–673.Google Scholar
[18] B., Fine, A. M., Gaglione and D., Spellman, Notions of discrimination, Comm. Algebra (2006), 2175–2182.Google Scholar
[19] B., Fine, A. M., Gaglione and D., Spellman, Discrimination properties in universal algebras, in preparation.
[20] G., Higman, Subgroups of finitely presented groups, Proc. Royal Soc. London Ser. A 264 (1961), 455–475.Google Scholar
[21] G., Higman, Finitely presented infinite simple groups, Notes on Pure Math. 8, (I.A.S., Austral. Nat. Univ., Canberra 1974).Google Scholar
[22] R., Hirshon and D., Meier, Groups with a quotient that contains the original group as a direct factor, Bull. Austral. Math. Soc. 45 (1992), 513–520.Google Scholar
[23] J. M., Tyrer Jones, Direct products and the Hopf property, J. Austral. Math. Soc. 17 (1974), 174–196.Google Scholar
[24] M., Kassabov, On discriminating solvable groups, preprint.
[25] O., Kharlampovich and A., Myasnikov, Irreducible affine varieties over a free group: I. Irreducibility of quadratic equations and Nullstellensatz, J. Algebra 200 (1998), 472–516.Google Scholar
[26] O., Kharlampovich and A., Myasnikov, Irreducible affine varieties over a free group: II. Systems in triangular quasi-quadratic form and a description of residually free groups, J. Algebra 200 (1998), 517–569.Google Scholar
[27] O., Kharlampovich and A., Myasnikov, Description of fully residually free groups and irreducible affine varieties over free groups, in Summer school in Group Theory in Banff, 1996, CRM Proc. Lecture Notes 17 (Amer. Math. Soc., Providence, RI 1999), 71–81.Google Scholar
[28] O., Kharlampovich and A., Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350(2) (1998), 571–613.Google Scholar
[29] O., Kharlampovich and A., Myasnikov, The elementary theory of free nonabelian groups, J. Algebra 302 (2006), 451–552.Google Scholar
[30] A. G., Kurosh, The Theory of Groups, Vol. I (Chelsea, New York 1956).Google Scholar
[31] A. G., Kurosh, The Theory of Groups, Vol. II (Chelsea, New York 1956).Google Scholar
[32] R. C., Lyndon, Problems in combinatorial group theory, in Annals of Mathematical Studies 111 (Princeton Univ. Press 1987), 3–33.Google Scholar
[33] W., Magnus, A., Karass and D., Solitar, Combinatorial Group Theory (Interscience, New York 1966).Google Scholar
[34] A. I., Mal'cev, On free soluble groups, Dokl. Akad. Nauk SSR 130 (1960), 495–498. English transl. Soviet Math. Dokl. (1960), 65–68.Google Scholar
[35] A. M., Myasnikov and P., Shumyatsky, Discriminating groups and c-dimension, J. Group Theory 7(1) (2004), 135–142.Google Scholar
[36] B. H., Neumann, Some remarks on infinite groups, J. London Math. Soc. 12 (1937), 120–127.Google Scholar
[37] H., Neumann, Varieties of Groups (Springer-Verlag, New York 1968).Google Scholar
[38] Z., Sela, Diophantine geometry over groups. I. Makanin-Razborov diagrams, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 31–105.Google Scholar
[39] Z., Sela, Diophantine geometry over groups. II. Completions, closures, and formal solutions, Israel J. Math. 104 (2003), 173–254.Google Scholar
[40] Z., Sela, Diophantine geometry over groups. III. Rigid and solid solutions, Israel J. Math. 147 (2005), 1–73.Google Scholar
[41] Z., Sela, Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence, Israel J. Math. 143 (2004), 1–130.Google Scholar
[42] Z., Sela, Diophantine geometry over groups. V1. Quantifier elimination, Israel J. Math. 150 (2005), 1–97.Google Scholar
[43] Z., Sela, Diophantine geometry over groups. VI. The elementary theory of a free group, Geom. Funct. Anal. 16 (2006), 707–730.Google Scholar
[44] W., Szmielew, Elementary properties of abelian groups, Fund. Math. 41 (1955), 203–271.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×