Centralizers of derived-from-Anosov systems on ${\mathbb{T}}^3$: rigidity versus triviality

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb{T}}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.