Realising end invariants by limits of minimally parabolic, geometrically finite groups

Abstract

We shall show that for a given homeomorphism type and a set of end invariants (including the parabolic locus) with necessary topological conditions which a topologically tame Kleinian group with that homeomorphism type must satisfy, there is an algebraic limit of minimally parabolic, geometrically finite Kleinian groups which has exactly that homeomorphism type and the given end invariants. This shows that the Bers–Sullivan–Thurston density conjecture follows from Marden’s conjecture proved by Agol and Calegari–Gabai combined with Thurston’s uniformisation theorem and the ending lamination conjecture proved by Minsky, partially collaborating with Masur, Brock and Canary.