arXiv subjects
Publications and source records attributed to Spyridon Petrakos.
We prove that any two rank-one systems are sub-$L^1$ orbit equivalent, fully clarifying quantitative orbit equivalence for a generic class of transformations. As a corollary, we establish unconditional optimality of Belinskaya's theorem.
We establish stable rank one for the reduced group C$^*$-algebras of the C$^*$-simple topological full groups and dynamical alternating groups constructed by Kerr and Tucker-Drob. The proof relies on both type II$_1$ and type III phenomena, in the first case via the use of Følner towers and in the second via Ozawa's recent results on selflessness as applied to direct products of free products.
We prove that any two $\mathbb{Z}$-odometers are sub-$L^1$-orbit equivalent, greatly strengthening previous results and giving a definitive picture of quantitative orbit equivalence for these systems.
We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces, and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.
We show that, for every minimal action of a countably infinite discrete group on a compact metrizable space, if the extreme boundary of the simplex of invariant Borel probability measures is closed and has finite covering dimension then the action has the small boundary property.
Given a topologically free action of a countably infinite amenable group on the Cantor set, we prove that, for every subgroup $G$ of the topological full group containing the alternating group, the group von Neumann algebra $\mathscr{L} G$ is a McDuff factor. This yields the first examples of nonamenable simple finitely generated groups $G$ for which $\mathscr{L} G$ is McDuff. Using the same construction we show moreover that if a faithful action $G\curvearrowright X$ of a countable group on a countable set with no finite orbits is amenable then the crossed product of the associated shift action over a given II$_1$ factor is a McDuff factor. In particular, if $H$ is a nontrivial countable ICC group and $G\curvearrowright X$ is a faithful amenable action of a countable ICC group on a countable set with no finite orbits, then the group von Neumann algebra of the generalized wreath product $H\wr_X G$ is a McDuff factor. Our technique can also be applied to show that if $H$ is a nontrivial countable group and $G\curvearrowright X$ is an amenable action of a countable group on a countable set with no finite orbits then the generalized wreath product $H\wr_X G$ is Jones-Schmidt stable.