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Description

This is joint work with Darren Long and Alan Reid. We show that for certain arithmetic Kleinian groups defined by quadratic forms, that geometrically finite subgroups are the intersection over all finite index subgroups containing them. These groups include the Bianchi groups and the isometry group of the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these, immersed incompressible surfaces lift to embeddings in a finite sheeted covering space. The method makes use of Coxeter groups, quadratic forms, and the four-square theorem.