Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

We describe homomorphisms $φ:H\rightarrow G$ 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 $φ$ is small or $φ$ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to $G$ as above. We prove a more precise result for homomorphisms $φ:H\rightarrow {\rm Mod}(Σ)$, where $H$ as above and ${\rm Mod}(Σ)$ is the mapping class group of a connected compact surface $Σ$. In this case there exists an open normal subgroup $V\leqslant H$ such that $φ(V)$ is finite. We also prove the analogous statement for homomorphisms $φ:H\rightarrow {\rm Out}(G)$, 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.

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