Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Abstract

We describe homomorphisms ϕ : H → 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 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 → Mod( ), where H is as above and Mod( ) is the mapping class group of a connected compact surface . In this case there exists an open normal subgroup V ≤ H such that ϕ(V ) is finite. We also prove the analogous statement for homomorphisms ϕ : H → 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.

Abstract group actions of locally compact groups on CAT(0) spaces

“…We will always assume that topological groups have the Hausdorff property. There are several results in this direction in the literature, see [5,15,17,18,23,27]. Here, the group G will be a locally compact group while H will be the isometry group of a CAT.0/ space equipped with the discrete topology.…”