  We prove that every countable family of countable acylindrically hyperbolic
groups has a common finitely generated acylindrically hyperbolic quotient. As
an application, we obtain an acylindrically hyperbolic group $Q$ with strong
fixed point properties: $Q$ has property $FL^p$ for all $p\in [1, +\infty)$,
and every action of $Q$ on a finite dimensional contractible topological space
has a fixed point. In addition, $Q$ has other properties which are rather
unusual for groups exhibiting "hyperbolic-like" behaviour. E.g., $Q$ is not
uniformly non-amenable and has finite generating sets with arbitrary large
balls consisting of torsion elements.
