  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 $\phi:\Gamma_1\to \Gamma_2$ 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 $\ell^1$-Novikov property for polynomially bounded homomorphisms of
groups with polynomially bounded cohomology in $\C$. 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.
