On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

Abstract

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.