Invariants of hyperbolic 3-manifolds in relative group homology

Let \(M\) be a complete oriented hyperbolic \(3\)--manifold of finite volume. Using classifying spaces for families of subgroups we construct a class \(\beta_P(M)\) in the Adamson relative homology group \(H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})\), where \(\bar{P}\) is the subgroup of parabolic transformations which fix \(\infty\) in the Riemann sphere. We also prove that the classes \(F(M)\) in the Takasu relative homology groups \(H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\) constructed by Zickert, which are not well-defined and depend of a choice of decorations by horospheres, are all mapped to \(\beta_P(M)\) via a canonical comparison homomorphism \(H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})\). To do this, we simplify the construction of the classes \(F(M)\) using a simpler complex which computes \(H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\), getting a simple simplicial formula for \(F(M)\), which in turn gives a simpler and more efficient formula to compute the volume and Chern--Simons invariant than the one given by Zickert. The constructions can be extended for any boundary-parabolic \(PSL_2(\mathbb{C})\)-representation.