Amenable actions, free products and a fixed point property Academic Article uri icon

abstract

  • We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit interesting actions of this kind. A complete characterization of such free products is given in terms of a fixed point property … In the early 20th century, the construction of Lebesgue's measure was followed by the discovery of the Banach–Hausdorff–Tarski paradoxes (see [21,19,4,32]; see also [17]). This prompted von Neumann [23] to study the following general question … Given a group G acting on a set X, when is there an invariant mean on X … If such a mean exists, the action is called amenable … For the study of the classical paradoxes, one also considers normalisations other than condition (ii) above … (2)] The notion of amenability later introduced by Zimmer [33] and its variants [3] are different, being in a sense dual to the above … The thrust of von Neumann's article was …

publication date

  • January 1, 2007