Bulletin of the London Mathematical Society Advance Access originally published online on April 25, 2008
Bulletin of the London Mathematical Society 2008 40(3):463-472; doi:10.1112/blms/bdn028
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© 2008 London Mathematical Society
On full groups of measure-preserving and ergodic transformations with uncountable cofinalities
Institut für Informatik
Universität Leipzig
D-04009 Leipzig
Germany
8323 Thunderhead Drive
Boulder, CO 80302
USA
chollan@bgnet.bgsu.edu
Institut für Informatik
Universität Leipzig
D-04009 Leipzig
Germany
ulbrich@informatik.uni-leipzig.de
Received 21 September 2007. Revision received 9 January 2008.
The group of all measure-preserving permutations of the unit interval and the full group of an ergodic transformation of the unit interval are shown to have uncountable cofinality and the Bergman property. Here, a group G is said to have the Bergman property if, for any generating subset E of G, some bounded power of E
E–1{1} already covers G. This property arose in a recent interesting paper of Bergman, where it was derived for the infinite symmetric groups. We give a general sufficient criterion for groups G to have the Bergman property. We show that the criterion applies to a range of other groups, including sufficiently transitive groups of measure-preserving, non-singular, or ergodic transformations of the reals; it also applies to large groups of homeomorphisms of the rationals, the irrationals, or the Cantor set.
2000 Mathematics Subject Classification 20F05, 28D15 (primary), 37A15, 22F10, 20B22 (secondary).