Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-22T19:05:49.912Z Has data issue: false hasContentIssue false
  • English
  • Français

A Descriptive View of Combinatorial Group Theory

Published online by Cambridge University Press:  15 January 2014

Simon Thomas
Simon Thomas*
Affiliation:
Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USAE-mail:sthomas@math.rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we will prove the inevitable non-uniformity of two constructions from combinatorial group theory related to the word problem for finitely generated groups and the Higman–Neumann–Neumann Embedding Theorem.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 17 , Issue 2 , June 2011 , pp. 252 - 264
Copyright
Copyright © Association for Symbolic Logic 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

REFERENCES

[1] Clemens, J., Gao, S., and Kecuris, A. S., Polish metric spaces: their classification and isometry groups, this Bulletin, vol. 7 (2001), pp. 361–375.Google Scholar
[2] Ferenczi, V., Louveau, A., and Rosendal, C., The complexity of classifying separable Banach spaces up to isomorphism, Journal of the London Mathematical Society, vol. 79 (2009), pp. 323–345.Google Scholar
[3] Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, The Journal of Symbolic Logic, vol. 54 (1989), pp. 894–914.Google Scholar
[4] Grigorchuk, R. I., Degrees of growth of finitely generated groups and the theory of invariant means, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, vol. 25 (1985), pp. 259–300.Google Scholar
[5] Higman, G., Neumann, B. H., and Neumann, H., Embedding theorems for groups, Journal of the London Mathematical Society, vol. 24 (1949), pp. 247–254.Google Scholar
[6] Hjorth, G., Orbit cardinals: on the effective cardinalities arising as quotient spaces of the form X/G where G acts on a Polish space X, Israel Journal of Mathematics, vol. 111 (1999), pp. 221–261.Google Scholar
[7] Hjorth, G. and Kecuris, A. S., Borel equivalence relations and classification of countable models, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 221–272.Google Scholar
[8] Hjorth, G. and Kecuris, A. S., The complexity of the classification of Riemann surfaces and complex manifolds, Illinois Journal of Mathematics, vol. 44 (2000), pp. 104–137.Google Scholar
[9] Jecu, T., Set theory, The third millennium ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[10] Kanovei, V., Borel equivalence relations: Structure and classification, University Lecture Series, vol. 44, American Mathematical Society, Providence, 2008.Google Scholar
[11] Kanovei, V. and Reeken, M., Some new results on the Borel irreducibility of equivalence relations, Izvestiya Mathematics, vol. 67 (2003), pp. 55–76.Google Scholar
[12] Kecuris, A. S., “AD + Uniformization” is equivalent to “Half ADℝ , Cabal seminar 81–85, Lecture Notes in Mathematics, vol. 1333, Springer, Berlin, 1988, pp. 98–102.Google Scholar
[13] Kecuris, A. S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, 1995.Google Scholar
[14] Martin, D. A., The axiom of determinacy and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.Google Scholar
[15] Martin, D. A., Borel determinacy, Annals of Mathematics, vol. 102 (1975), pp. 363–371.Google Scholar
[16] Martin, D. A. and Solovay, R. M., A basis theorem for sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–159.Google Scholar
[17] Neumann, B. H., Some remarks on infinite groups, Journal of the London Mathematical Society, vol. 12 (1937), pp. 120–127.Google Scholar
[18] Sacks, G. E., Forcing with perfect closed sets, Axiomatic set theory, part I, Proceedings of Symposia in Pure Mathematics, vol. XIII, American Mathematical Society, Providence, RI, 1971, pp. 331–355.Google Scholar
[19] Snoenfield, J. R., The problem of predicativity, Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, Jerusalem, 1961.Google Scholar
[20] Thomas, S., The classification problem for torsion-free abelian groups of finite rank, Journal of the American Mathematical Society, vol. 16 (2003), pp. 233–258.Google Scholar
[21] Thomas, S., A remark on the Higman–Neumann–Neumann embedding theorem, Journal of Group Theory, vol. 12 (2009), pp. 561–565.Google Scholar
[22] Thomas, S. and Velickovic, B., On the complexity of the isomorphism relation for finitely generated groups, Journal of Algebra, vol. 217 (1999), pp. 352–373.Google Scholar