Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF

Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF

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Hongbin Sun , Rutgers University
Fine Hall 314

A group G is called LERF if the property that an element not lying in a finitely generated subgroup is visible via a finite quotient of G. LERFness of groups is closed related with low-dimensional topology: whether an immersed \pi_1-injective object can be lifted to embedding in some finite cover. We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.