Wreath products of groups acting with bounded orbits

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