Wreath products of groups acting with bounded orbits

Paul-Henry Leemann, Grégoire Schneeberger

Published 2021-02-16Version 1

If S is a structure over (a concrete category over) metric spaces, we say that a group G has property BS if any action on a S-space has bounded orbits. Examples of such structures include metric spaces, Hilbert spaces, CAT(0) cube complexes, connected median graphs, trees or ultra-metric spaces. The corresponding properties BS 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 (cof $\neq\omega$). Our main result is that for a large class of structures S, the wreath product $G\wr_XH$ has property BS if and only if both $G$ and $H$ have property BS 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 cof $\neq\omega$ if and only if both $G$ and $H$ have cof $\neq\omega$ and $H$ acts on $X$ with finitely many orbits.