Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups · arXivDesk
1907.10166Jul 23, 201935 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
Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups
We describe homomorphisms φ:H→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 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(Σ)
Nearby in the stack
, where
H
as above and
Mod(Σ)
is the mapping class group of a connected compact surface
Σ
. 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.