# Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

## TL;DR

This paper characterizes homomorphisms from certain topological groups to acylindrically hyperbolic groups, revealing conditions under which these maps are almost continuous or have small images, with applications to specific groups like the Hawaiian earring group.

## Contribution

It provides new insights into the structure of homomorphisms from topological groups to acylindrically hyperbolic groups, including automatic continuity results and specific cases involving mapping class groups and hyperbolic groups.

## Key findings

- Homomorphisms are either nearly continuous or have small images.
- Hawaiian earring group is acylindrically hyperbolic but lacks a universal acylindrical action.
- Existence of finite images for certain homomorphisms to mapping class groups.

## Abstract

We describe homomorphisms $\varphi:H\rightarrow G$ 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 $\varphi$ 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 $\varphi:H\rightarrow {\rm Mod}(\Sigma)$, where $H$ as above and ${\rm Mod}(\Sigma)$ 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 $\varphi:H\rightarrow {\rm Out}(G)$, 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.