The Strong Independence Theorem for Automorphism Groups and Congruence Lattices of Arbitrary Lattices

In the book, General Lattice Theory, the first author raised the following problem (Problem II.18): Let L be a nontrivial lattice and let G be a group. Does there exist a lattice K such that K and L have isomorphic congruence lattices and the automorphism group of K is isomorphic to G? The finite case was solved, in the affirmative, by V.A. Baranskĭı and A. Urquhart in 1978, independently. In 1995, the first author and E.T. Schmidt proved a much stronger result, the strong independence of the automorphism group and the congruence lattice in the finite case. In this paper, we provide a full affirmative solution of the above problem. In fact, we prove much stronger results, verifying strong independence for general lattices and also for lattices with zero.