Non-cancellative varieties, maximal tori, and the Makar-Limanov invariant

In this paper, we construct a wide class of new counterexamples to the generalized Zariski cancellation problem.

The cylinders over these counterexamples have infinitely many non-conjugate maximal tori in their regular automorphism groups.

We provide an example of a variety with maximal tori of different dimensions in its automorphism group.

Additionally, we prove that a cylinder over a non-rigid trinomial variety without a line factor is generically flexible, and hence, has a trivial Makar-Limanov invariant.