We prove that, for any two finite volume hyperbolic 3-manifolds, the amalgamation of their fundamental groups along any nontrivial infinite index geometrically finite subgroup is not LERF (locally extended residually finite). This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic 3-manifold groups along abelian subgroups. A consequence of this result is that, for all closed arithmetic hyperbolic 4-manifolds have nonLERF fundamental groups. Along with the author's previous work, we obtain that, for any arithmetic hyperbolic manifold with dimension at least 4, with possible exceptions in seven-dimensional manifolds defined by the octonion, its fundamental group is not LERF.
All Science Journal Classification (ASJC) codes
- 57M05 (primary)
- 57M50 (secondary)