Replacing virtual centralizer equality by intrinsic largeness

Establish centralizer rigidity for perturbations of non-trivial diagonal maps under an intrinsic largeness condition on the perturbed centralizer, such as a dimension-count assumption or the absence of rank-one factors, instead of assuming that the perturbed centralizer is virtually isomorphic to the algebraic centralizer.

Background

The main theorem assumes virtual isomorphism between the smooth centralizer of the perturbation and the algebraic centralizer of the diagonal model. This algebraic comparison is used to identify the relevant Lie algebras and construct partially hyperbolic centralizer elements.

The authors ask whether virtual equality can be replaced by a more intrinsic condition expressing that the centralizer is sufficiently large. They specifically mention dimension counting, as in the generic case, and a no-rank-one-factor condition related to a conjecture of Damjanović–Wilkinson–Wu–Xu.