On the simplicity of the full group of ergodic transformations

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.

Section: Measure-preserving Transformations Of [0 1]supporting

confidence: 55%

“…By (7) we find some h ∈ G(Σ) mapping Δ = Δf onto Δ. Hence h = (fh) Δ ∪ id Ω\Δ ∈ G(Δ) by (8). With (2) and Δ Σ we have G(Δ) ⊆ G(Σ).…”

Section: Strong Cofinality and The Bergman Propertymentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Droste

Holland

Ulbrich

2008

“…To the best of our knowledge, there is still no complete algebraic description of full groups. However, some partial results clarifying how different dynamical properties affect the algebraic structure of the full group have been earlier established; see, for example, [4,8,12].…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of orbits of Cantor systems from full groups

Medynets

2011

In this note, we show that the topological structure of orbits of a Cantor dynamical system, satisfying some mild technical conditions, is completely determined by its full group; namely, we prove that two Cantor systems (X1, G1) and (X2, G2) are orbit equivalent if and only if their full groups are isomorphic as abstract groups. Furthermore, any isomorphism between full groups is generated by a homeomorphism of the underlying spaces. This result is a complete topological analog of the well‐known Dye's theorem proved originally for ergodic measure‐preserving actions. Our results are established for systems having no orbits of length 1 and 2 and such that every open set is intersected by some orbit at least twice.

“…Full groups of type III ergodic automorphisms are contractible, as was shown by Danilenko [3]. Besides, Eigen [4] has proved that they are simple. Therefore, the preceding theorem provides examples of simple contractible Polish groups with ample generics, as opposed to the highly non-simple groups we obtain with Theorem B.…”

mentioning

confidence: 73%

Connected Polish groups with ample generics

Kaïchouh

Maître²

2015

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.