Strongly chiral rational homology spheres with hyperbolic fundamental groups

For each $m\geq0$ and any prime $p\equiv3\ \mathrm{(mod \ 4)}$, we construct strongly chiral rational homology $(4m+3)$-spheres, which have real hyperbolic fundamental groups and only non-zero integral intermediate homology groups isomorphic to $\mathbb{Z}_{2p}$ in degrees $1,2m+1$ and $4m+1$. This gives group theoretic analogues in high dimensions of the existence of strongly chiral hyperbolic rational homology $3$-spheres, as well as of the existence of strongly chiral hyperbolic manifolds of any dimension that are not rational homology spheres, which was shown by Weinberger. One of our tools will be $r$-spins. We thus investigate the relationship between the sets of degrees of self-maps of a given manifold and its $r$-spins, and give classes of manifolds for which the sets are equal.