No CrossRef data available.
Article contents
A Descriptive View of Combinatorial Group Theory
Published online by Cambridge University Press: 15 January 2014
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
- Copyright
- Copyright © Association for Symbolic Logic 2011
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