Skip to main content Accessibility help
×
Home

UNIVERSAL COUNTABLE BOREL QUASI-ORDERS

  • JAY WILLIAMS (a1)

Abstract

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders.

In this paper we are concerned with universal countable Borel quasi-orders, i.e., countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We first establish that there is a universal countable Borel quasi-order, and then establish that several countable Borel quasi-orders are universal. An important example is an embeddability relation on descriptive set theoretic trees.

Our main result states that embeddability of finitely generated groups is a universal countable Borel quasi-order, answering a question of Louveau and Rosendal. This immediately implies that biembeddability of finitely generated groups is a universal countable Borel equivalence relation. The same techniques are also used to show that embeddability of countable groups is a universal analytic quasi-order.

Finally, we show that, up to Borel bireducibility, there are ${2^{{\aleph _0}}}$ distinct countable Borel quasi-orders, which symmetrize to a universal countable Borel equivalence relation.

Copyright

References

Hide All
[1]Adams, Scot and Kechris, Alexander S., Linear algebraic groups and countable Borel equivalence relations. Journal of American Mathematical Society, vol. 13 (2000), no. 4, pp. 909943.
[2]Champetier, Christophe, L’espace des groupes de type fini. Topology, vol. 39 (2000), no. 4, pp. 657680.
[3]Dougherty, R., Jackson, S., and Kechris, A. S., The structure of hyperfinite Borel equivalence relations. Transactions of the American Mathematical Society, vol. 341 (1994), no. 1, pp. 193225.
[4]Dougherty, Randall and Kechris, Alexander S., How many Turing degrees are there? Computability theory and its applications (Boulder, CO, 1999), Contemporary Mathematics, vol. 257, American Mathematical Society, Providence, RI, 2000, pp. 8394.
[5]Feldman, Jacob and Moore, Calvin C., Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Transactions of the American Mathematical Society, vol. 234 (1977), no. 2, pp. 289324
[6]Friedman, Sy-David and Ros, Luca Motto, Analytic equivalence relations and bi-embeddability. this Journal, vol. 76 (2011), no. 1, pp. 243–266.
[7]Gao, Su, Coding subset shift by subgroup conjugacy. Bulletin of the London Mathematical Society, vol. 32 (2000), no. 6, pp. 653657.
[8]Kechris, Alexander S., Classical descriptive set theorzy. Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[9]Louveau, Alain and Rosendal, Christian, Complete analytic equivalence relations. Transactions of the American Mathematical Society, vol. 357 (2005), no. 12, pp. 48394866.
[10]Lyndon, Roger C. and Schupp, Paul E., Combinatorial group theory. Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition.
[11]Mekler, Alan H., Stability of nilpotent groups of class 2 and prime exponent. this Journal, vol. 46 (1981), no. 4, pp. 781788.
[12]Thomas, Simon and Velickovic, Boban, On the complexity of the isomorphism relation for finitely generated groups. Journal of Algebra, vol. 217 (1999), no. 1, pp. 352373.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed