On profinite rigidity, Grothendieck pairs, and the second homology of some $3$-orbifold groups

Abstract

The second homology group is of central importance in the study of profinite rigidity of $3$-manifold groups. Although general and deep results imply that the integral homology of cocompact hyperbolic $3$-orbifold groups is computable in principle, the resulting algorithm is not practical. We develop an effective method for computing $H_2$ in the case of orbifold groups arising as finite extensions of the fundamental group of hyperbolic rational homology $3$-spheres. As a special case, this yields explicit computations of the second homology groups of all cocompact lattices between $\pi_1(\mathcal{W})$ and its normalizer in $\mathrm{PSL}_2(\mathbb C)$, where $\mathcal{W}$ is the Weeks manifold. We also show that these lattices are absolutely profinitely rigid, completing work by Bridson, McReynolds, Reid & Spitler in this setting. As a special case, we determine that $H_2(\Gamma_{\mathcal{O}}, \mathbb Z) \cong \mathbb{Z} / 2\mathbb{Z}$, where $\Gamma_{\mathcal{O}}$ is the normalizer of the group of units $\Gamma_{\mathcal{O}}^1$ in a choice of maximal order $\mathcal{O}$ of the quaternion algebra associated to $\mathcal{W}$, thereby answering a question of Bridson & Reid. Although this non-vanishing obstructs one possible construction of Grothendieck pairs in $\Gamma_{\mathcal{O}}^1 \times \Gamma_{\mathcal{O}}^1$, we use our computations to show the vanishing of the second homology of another lattice whose derived subgroup is $\Gamma_{\mathcal{O}}^1$, which then yields Grothendieck pairs in this direct product by a theorem of Bridson & Reid. Finally, to showcase the generality of the techniques, we also compute the second homology of some finite extensions by orientable isometries of the fundamental group of some Fibonacci manifolds $M_n$.