Wreath products of groups acting with bounded orbits
Abstract
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 Sspace has bounded orbits. Examples of such structures include metric spaces, Hilbert spaces, CAT(0) cube complexes, connected median graphs, trees or ultrametric 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.08001
 Bibcode:
 2021arXiv210208001L
 Keywords:

 Mathematics  Group Theory;
 20E22;
 20F65
 EPrint:
 17 pages