Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Abstract

We describe homomorphisms $$\varphi :H\rightarrow G$$ for which G is acylindrically hyperbolic and H 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 $$\varphi :H\rightarrow {\text {Mod}}(\Sigma )$$ , where H is as above and $${\text {Mod}}(\Sigma )$$ 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 $$\varphi :H\rightarrow {\text {Out}}(G)$$ , 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.