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The class A of countable groups that admit a faithful, transitive, amenable-- in the sense that there is an invariant mean -- action on a set has beenwidely investigated in the past. In this paper, we no longer require the actionto be transitive, but we ask for it to preserve a locally finite metric (andstill to be faithful and amenable). The groups having such actions are thosethat embed into a totally disconnected amenable locally compact group. Then wefocus on the subclass A 1 of groups for which the actions are moreovertransitive. This class is strictly contained into A and includes non-amenablegroups. An important particular case of actions preserving a locally finitemetric is given by actions by automorphisms of locally finite connected graphs.We take this opportunity, in our partly expository paper, to review some niceresults about amenable actions in this setting.

Amenable actions preserving a locally finite metric