Wreath products of groups acting with bounded orbits

Abstract

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, $\mathrm{CAT}(0)$ 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 $\text{(T)}$), property $\mathrm{FW}$ (both for $\mathrm{CAT}(0)$ 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.