Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Abstract: We describe homomorphisms $\varphi:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of $\varphi$ is small or $\varphi$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to $G$ as above. We prove a more precise result for homomorphisms $\varphi:H\rightarrow {\rm Mod}(\Sigma)$, where $H$ as above and ${\rm Mod}(\Sigma)$ is the mapping class group of a connected compact surface $\Sigma$. In this case there exists an open normal subgroup $V\leqslant H$ such that $\varphi(V)$ is finite. We also prove the analogous statement for homomorphisms $\varphi:H\rightarrow {\rm Out}(G)$, where $G$ is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.