Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

Groups with infinite FC-center have the Schmidt property

We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

Zero-dimensional and symbolic extensions of topological flows

<p style='text-indent:20px;'>A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [<xref ref-type="bibr" rid="b6">6</xref>] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-<inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> map admits an extension by a subshift for any <inline-formula><tex-math id="M2">\begin{document}$ t\neq 0 $\end{document}</tex-math></inline-formula>. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on <inline-formula><tex-math id="M3">\begin{document}$ \{0,1\}^{\mathbb Z} $\end{document}</tex-math></inline-formula> with a roof function <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> vanishing at the zero sequence <inline-formula><tex-math id="M5">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula> admits a principal symbolic extension or not depending on the smoothness of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M7">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula>.</p>

Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

<p style='text-indent:20px;'>We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.</p>

Unicorn paths and hyperfiniteness for the mapping class group

Let S be an orientable surface of finite type. Using Pho-on’s infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc graph and the curve graph of S. In the curve graph case, this strengthens the results of Hamenstädt and Kida that this action is universally amenable and that the mapping class group of S is exact.

On Polish groups admitting non-essentially countable actions

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.