Central extensions and proper actions on 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. This is achieved by using quasimorphisms as a bridge between central extensions and group actions. As applications, we show that mapping class groups of finite-type surfaces possibly with boundary, multicurve stabilizers, and outer automorphism groups of torsion-free one-ended hyperbolic groups 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 analogue of property QT related to locally uniform exponential growth of groups, and derive the same stability results under central extensions. We provide several examples with or without property PH. In particular, the fundamental group of a compact orientable $3$-manifold $M$ has property PH whenever no summand in the sphere-disk decomposition of $M$ supports Nil geometry.