The algebraic cheap rebuilding property

We present an axiomatic approach to combination theorems for various homological properties of groups and, more generally, of chain complexes. Examples of such properties include algebraic finiteness properties, $\ell^2$-invisibility, $\ell^2$-acyclicity, lower bounds for Novikov--Shubin invariants, and vanishing of homology growth. As a key example, we introduce an algebraic version of Ab\'ert--Bergeron--Fr\k{a}czyk--Gaboriau's cheap rebuilding property that implies vanishing of torsion homology growth and fits into our axiomatic framework for combination theorems. In particular, we obtain that certain graphs of groups with amenable vertex groups and elementary amenable edge groups have vanishing torsion homology growth.