Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property

We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $\varphi :{\mathrm{\Gamma }}_1\to {\mathrm{\Gamma }}_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong ${\mathrm{\ell }}^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $\text{C}$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $\pi_1(\partial W)\to \pi_1(W)$ belonging to the class above, are homotopy invariant.