Positive scalar curvature and low-degree group homology

Let $\Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} \Gamma$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $\Gamma$ with coefficients in the $\mathbb{C}\Gamma$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.