C. R. Acad. Sci. Paris, t. 324, Sbrie I, p. 1255-1258, 1997 TopologielTopology Moyennabiliti des groupes dhnombrables et actions sur les espaces de Cantor Thierry GIORDANO et Pierre de la HARPE T. G. : IXpartemmt dr Mathlmatiqurs et Statistiques, tJniversitt d’ottawa? Ottawa KlN 6N5, Canada. E-mail : &iordanoQmatrix.~c..uottawa.c:a P. H. : Section de Mathimatiqurs. C.P. 240, CH-1211 Gem% 24, Suisse. E-mail : Pierrr.ddaHarpc@math.unigr.c~h R&urn& Nous rCpondons a une question de R. Grigorchuk en montrant la caractCrisation suivante : pour qu’un groupe dknombrable r soit moyennable, ii faut et il suffit que toute action continue de I? sur l’ensemhle de Cantor possi?de une mesure de probabilitCinvariante. La dkmonstration utilise une variante facile d’un r&ultat classique d’Alexandroff et Urysohn : tout r-espacecompact mCtrisable est une image surjective Cquivariante d’un r-espace de Cantor. Amenability of countable groups and actions on Cantor sets Abstract. We answer a question of R. Grigorchukby .showing thefollowing characterization: for a countable group r to be amenable, it is necessary and s@icient that any continuous action $l‘ on the Cantor sethasan invariant probability meusure. Theproof uses an easy variation qf a classical result of Alexandroff and Urysohn: any metrisable r-space is an equivariant sutjective image qf a F-Cantor set. Abridged English Version Any metrizable compact space is a continuous image of the Cantor set K: it is a well known result due to Alexandroff and Urysohn. The first goal of this Note is to observe the following equivariant formulation of the Alexandroff-Urysohn result: PROPOSITION. - Let r be a countable group, X a metrizable compact space, and N : r‘ x X + X a continuous action. Then, there exists a continuous action & : r x K ---+ K on the Cantor set K and a continuous mapping x : K -+ X which is equivariant and onto. !f; moreover, X is an injinite set and if the action o is minimal, then there exists G and x as above with & minimal. There are many equivalent definitions of amenability for a group (seee.g. [7], [8], and [13]). For a countable group, let us recall the one going back to Bogolyubov (see [3] and [5]): I? is amenable (f, Note prksent6e par ktienne WHYS. 0764.4442/Y7/03241255 0 AcadCmie des Sciences/Elsevier, Paris 1255