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• W3213516837 abstract "We describe homomorphisms $$varphi :Hrightarrow G$$ 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 $$varphi :Hrightarrow {text {Mod}}(Sigma )$$ , 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 $$Vleqslant H$$ such that $$varphi (V)$$ is finite. We also prove the analogous statement for homomorphisms $$varphi :Hrightarrow {text {Out}}(G)$$ , 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." @default.
• W3213516837 created "2021-11-22" @default.
• W3213516837 creator A5011460177 @default.
• W3213516837 creator A5064620696 @default.
• W3213516837 date "2021-11-15" @default.
• W3213516837 modified "2023-10-14" @default.
• W3213516837 title "Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups" @default.
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• W3213516837 doi "https://doi.org/10.1007/s00208-021-02278-4" @default.
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