On the profinite distinguishability of hyperbolicDehn fillings of finite-volume 3–manifolds

We use the Culler-Shalen machine and tools from model theory to study the profinite rigidity of residually finite groups, especially 3-manifold groups.We borrow a transfer principle from model theory to apply to C-character varieties in order to study cofinite collections of F p -character varieties and prove that under certain finiteness conditions weaker than non-Hakenness, they all have the same (finite) cardinality.We prove that residually finite groups satisfying a niceness property are almost relatively profinitely distinguishable within a geometrically relevant class, and we finish up by applying that result to knot complements in S 3 in particular.20F65, 57M07; 03C07, 03C52