Tangent bundles of hyperbolic spaces and proper affine actions on $L^p$ spaces

We define the notion of a negatively curved tangent bundle of a metric measured space. We prove that, when a group $G$ acts on a metric measured space $X$ with a negatively curved tangent bundle, then $G$ acts on some $L^p$ space, and that this action is proper under suitable assumptions. We then check that this result applies to the case when $X$ is a hyperbolic space.

Property $P_{naive}$ for acylindrically hyperbolic groups

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, dots, h_k$, there exists another element $gamma eq 1$ such that for all $i$, $langle h_i, gamma rangle = langle h_i rangle* langle gamma rangle$. We also obtain that if a collection of subgroups $H_1, dots, H_k$ is a hyperbolically embedded collection, then there is $gamma eq 1$ such that for all $i$, $langle H_i, gamma rangle = H_i * langle gamma rangle$.