The homotopy type of a finite 2-complex with non-minimal Euler characteristic

We resolve two long-standing and closely related problems concerning stably free $mathbb{Z} G$-modules and the homotopy type of finite 2-complexes. In particular, for all $k ge 1$, we show that there exists a group $G$ and a non-free stably free $mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k ge 0$, there exists homotopically distinct finite 2-complexes with fundamental group $G$ and with Euler characteristic $k$ greater than the minimal value over $G$. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.