Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic \(n\)-orbifolds (Q2766432)

The author proves that, if the group \(G\) is the fundamental group of a Haken 3-manifold or the fundamental orbifold group of a closed hyperbolic orbifold of any dimension, then the Abelian subgroups of \(G\) are separable in \(G\); that is, given any Abelian subgroup \(H\) of \(G\) and given any element \(g\in G\setminus H\), there is a subgroup \(K\) of \(G\) which has finite index in \(G\), and which contains \(H\) but does not contain \(g\).NEWLINENEWLINENEWLINEIn the 3-manifold case, the proof follows the outline of the proof that groups of Haken 3-manifolds are residually finite: (1) reduce to the case of a closed Haken 3-manifold; (2) apply the Jaco-Shalen-Johannson decomposition theorem; (3) apply known results for Seifert-fibered spaces and hyperbolic 3-manifolds; (4) carefully piece these results together.NEWLINENEWLINENEWLINEIn the general hyperbolic case, apply Selberg's lemma, Mostow rigidity, and analyse the trace field associated with a representation from \(G\) into \(\text{SO}(1,n; R)\) by matrix and number theoretic methods.