arXiv subjects

Grégoire Schneeberger

Publications and source records attributed to Grégoire Schneeberger.

4 recordsLinked to original sources

arXiv2024

Proper actions of Grigorchuk groups on a CAT(0) cube complex

On this paper we will present a construction of a CAT(0) cube complex (an infinite cube), on which the uncountable family of Grigorchuk groups $G_ω$ act without bounded orbit. Moreover, if the sequence $ω$ does not contain repetition, we prove that the action is proper and faithful. As a consequence of this result, this cube complex is a model for the classifying space of proper actions for all the groups $G_ω$ with $ω$ without repetition. This construction works in a general way for any group acting on a set and which admits a commensurated subset.These examples of non-elliptic actions of infinite finitely generated torsion groups on a non-positively curved cube complex contrast to several established fixed-point theorems concerning actions of torsion groups.

math.GR↗

arXiv2024

Wreath products of groups acting with bounded orbits

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.

math.GR↗

arXiv2022

Property FW and wreath products of groups: a simple approach using Schreier graphs

The group property FW stands in-between the celebrated Kazdhan's property (T) and Serre's property FA. Among many characterizations, it might be defined, for finitely generated groups, as having all Schreier graphs one-ended. It follows from the work of Y. Cornulier that a finitely generated wreath product $G\wr_XH$ has property~FW if and only if both $G$ and $H$ have property FW and $X$ is finite. The aim of this paper is to give an elementary, direct and explicit proof of this fact using Schreier graphs.

math.GR↗

arXiv2022

A Cheeger-Buser-Type inequality on CW complexes

We extend the definition of boundary expansion to CW complexes and prove a Cheeger-Buser-type relation between the spectral gap of the Laplacian and the boundary expansion of an orientable CW complex.

math.CO↗