  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.
