Filling 1-cycles, 2-cycle complexity, and torsion growth

For 2-complexes with trivial first Betti number, we study quantitative connections between filling inequalities for 1-cycles, the complexity of the second homology, and the size of the torsion part of the first homology. For 2-complexes whose fundamental groups have property $(\tau )$ with respect to all finite index normal subgroups, we prove a geometric lower bound on the logarithm of the size of the first homology of finite covers.