Proper actions on finite products of hyperbolic spaces

The main result of this paper identifies boundedness of the Euler class as the exact obstruction to preserving property QT under central extensions. For a central extension of groups $1\to Z\to E\to G\to 1$, we prove that $E$ has property QT if and only if $Z$ is finitely generated, $G$ has property QT, and the Euler class of the extension is bounded. These are done by using quasimorphisms as a bridge between central extensions and group actions. As applications, we show that the mapping class group of any finite-type surface possibly with boundary and the outer automorphism group of any torsion-free one-ended hyperbolic group have property QT. We also show that Sela's central extension description of the latter has a bounded Euler class.
In addition, we introduce property PH, which is a weaker analogy of property QT related to locally uniform exponential growth of groups, and derive the same stability results under central extensions. We provide a bunch of examples with or without property PH. In particular, the fundamental group of a compact orientable $3$-manifold $M$ has property PH if no summand in the sphere-disk decomposition of $M$ supports Nil geometry.