Understanding how a group can act on a given type of space can be a valuable tool for proving properties of both the group and the space. This thesis focuses on three distinct topics involving actions of infinite groups on graphs, cube complexes and metric spaces.

In the first part, we answer a question by Grigorchuk asking whether it is possible to give an explicit and elementary description of a CAT(0) cube complex on which the groups he defined act and, if so, to describe these ac- tions. In the following, we present the results of joint projects with P-H. Leemann where we are interested in the stability of certain properties of groups for the wreath product. Then, we study the stability of these properties for the wreath product in a more general framework. Finally, we study the notion of expansion for objects of dimension greater than 1.