Invariants of hyperbolic 3-manifolds in relative group homology

Abstract

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.