arXiv subjects
Publications and source records attributed to Sylvain Arnt.
Using the work of Cornulier-Valette and Whyte (see Appendix A for the latter), we show that neither the Haagerup property nor weak amenability is invariant under quasi-isometry of finitely generated groups. Appendix B shows, using the same examples, that the same holds for vanishing of the equivariant $L^p$-compression.
We define the structure of spaces with labelled partitions which generalizes the structure of spaces with measured walls and study the link between actions by automorphisms on spaces with labelled partitions and isometric affine actions on Banach spaces, and more particularly, on $L^p$ spaces. We build natural spaces with labelled partitions for the action of various contructions of groups, namely: direct sum; semi-direct product; wreath product and amalgamated free product. We apply this to prove that the wreath product of a group with property $PL^p$ by a group with Haagerup property has property $PL^p$ and the amalgamated free product over finite subgroups of groups with property $PL^p$ has property $PL^p$.
We show the necessary part of the following theorem : a finitely generated, residually finite group has property $PL^p$ (i.e. it admits a proper isometric affine action on some $L^p$ space) if, and only if, one (or equivalently, all) of its box spaces admits a fibred coarse embedding into some $L^p$ space. We also prove that coarse embeddability of a box space of a group into a $L^p$ space implies property $PL^p$ for this group.