Homology torsion growth in polynomially-growing mapping tori

We show that mapping tori of polynomially-growing automorphisms of many non-positively curved groups have vanishing homology torsion growth. In the case of free-by-cyclic groups with polynomially-growing monodromy, this confirms a conjecture of Lück relating the integral torsion and the $L^2$-torsion of a group. Our main tool is the cheap rebuilding property introduced by Abert—Bergeron—Fraczyk—Gaboriau. This is based on joint work with Naomi Andrew, Yassine Guerch and Sam Hughes.