Methods to compute ring invariants and applications: a new class of exotic threefolds

We develop some methods to compute the Makar-Limanov and Derksen invariants, isomorphism classes and automorphism groups for k-domains B, which are constructed from certain Russell k-domains. We propose tools and techniques to distinguish between k-domains with the same Makar-Limanov and Derksen invariants. In particular, we introduce the exponential chain associated to certain modifications. We extract C-domains from the class B that have smooth contractible factorial Spec(B), which are diffeomorphic to R 6 but not isomorphic to C 3, that is, exotic C 3. We examine associated exponential chains to prove that exotic threefolds Spec(B) are not isomorphic to Spec(R), for any Russell C-domain R.