Wreath products of groups acting with bounded orbits

If \mathbf{S} is a subcategory of metric spaces, we say that a group G has property \mathrm{B}\mathbf{S} if any isometric action on an \mathbf{S} -space has bounded orbits. Examples of such subcategories include metric spaces, affine real Hilbert spaces, \operatorname{CAT}(0) cube complexes, connected median graphs, trees or ultra-metric spaces. The corresponding properties {\mathrm{B}\mathbf{S}} are respectively Bergman’s property, property \mathrm{FH} (which, for countable groups, is equivalent to the celebrated Kazhdan’s property \textup{(T)} ), property \mathrm{FW} (both for \operatorname{CAT}(0) cube complexes and for connected median graphs), property \mathrm{FA} and uncountable cofinality. Historically many of these properties were defined using the existence of fixed points. Our main result is that for many subcategories \mathbf{S} , the wreath product G\wr_{X}H has property {\mathrm{B}\mathbf{S}} if and only if both G and H have property {\mathrm{B}\mathbf{S}} and X is finite. On one hand, this encompasses in a general setting previously known results for properties \mathrm{FH} and \mathrm{FW} . On the other hand, this also applies to the Bergman’s property. Finally, we also obtain that G\wr_{X}H has uncountable cofinality if and only if both G and H have uncountable cofinality and H acts on X with finitely many orbits.