I will present an axiomatic combination theorem that applies to several
properties of groups, such as: finiteness properties, vanishing of
$\ell^2$-Betti numbers, vanishing of $\mathbb{F}_p$-homology growth, and the
algebraic cheap rebuilding property. The latter implies vanishing of torsion
homology growth and is satisfied by elementary amenable groups. Joint with
Clara Löh, Marco Moraschini, Roman Sauer, and Matthias Uschold.