Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

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