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We survey the recent developments concerning fixed point properties for group actions on Banach spaces. In the setting of Hilbert spaces such fixed point properties correspond to Kazhdan's property (T). Here we focus on the general, non-Hilbert case, we discuss the methods, examples and several applications.

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}\mathrm{P}^{m-1}$ and $\mathbb{H}\mathrm{P}^{m-1}$.

We prove that for any countable acylidrically hyperbolic group $G$, there exists a generating set $S$ of $G$ such that the corresponding Cayley graph $Γ(G,S)$ is hyperbolic, $|\partialΓ(G,X)|>2$, the natural action of $G$ on $Γ(G,S)$ is acylindrical, and the natural action of $G$ on the Gromov boundary $\partialΓ(G,S)$ is hyperfinite. This result broadens a class of groups that admit a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.

A lot of what I do as rOpenSci's community assistant is related to social media.I check for posts about rOpenSci packages, invite people to share usecases,advertise upcoming events, as well as promotingnew packages which completed software peer review.