Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Abstract

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

35 pages, 2 figures. This version contains stronger theorems A and B and new theorems C and D about the mapping class groups and the outer automorphism groups of one-ended hyperbolic groups