Towers of regular self-covers and linear endomorphisms of tori

Abstract

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 iterates $p^n:M\to M$ are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that ${\pi }_1\left(M\right)$ 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 Kähler and $p$ is a strongly regular, holomorphic self-cover, and prove that a finite cover of $M$ splits as a product with a torus factor.