Wreath products of groups acting with bounded orbits

  • Paul-Henry Leemann

    Université de Neuchâtel, Neuchâtel, Switzerland
  • Grégoire Schneeberger

    Université de Genève, Genève, Switzerland
Wreath products of groups acting with bounded orbits cover
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Abstract

If is a subcategory of metric spaces, we say that a group has property  if any isometric action on an -space has bounded orbits. Examples of such subcategories include metric spaces, affine real Hilbert spaces, cube complexes, connected median graphs, trees or ultra-metric spaces. The corresponding properties are respectively Bergman’s property, property  (which, for countable groups, is equivalent to the celebrated Kazhdan’s property ), property  (both for cube complexes and for connected median graphs), property  and uncountable cofinality. Historically many of these properties were defined using the existence of fixed points. Our main result is that for many subcategories , the wreath product has property  if and only if both and have property  and is finite. On one hand, this encompasses in a general setting previously known results for properties and . On the other hand, this also applies to the Bergman’s property. Finally, we also obtain that has uncountable cofinality if and only if both and have uncountable cofinality and acts on with finitely many orbits.

Cite this article

Paul-Henry Leemann, Grégoire Schneeberger, Wreath products of groups acting with bounded orbits. Enseign. Math. 70 (2024), no. 1/2, pp. 121–149

DOI 10.4171/LEM/1059