Towers of regular self-covers and linear endomorphisms of tori

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $\pi_1(M)$ surjects onto a nontrivial free abelian group $A$, and the self-cover is induced by a linear endomorphism of $A$. Under further hypotheses we show that a finite cover of $M$ admits the structure of a principal torus bundle. We show that this applies when $M$ is K\"ahler and $p$ is a strongly regular, holomorphic self-cover, and prove that a finite cover splits as a product with a torus factor.