Towers of regular self-covers and linear endomorphisms of tori

van Limbeek, Wouter¹

¹ Department of Mathematics, University of Michigan, Ann Arbor, MI, United States

Geometry & topology, Tome 22 (2018) no. 4, pp. 2427-2464.

Résumé

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.

Détail

DOI : 10.2140/gt.2018.22.2427

Classification : 57N99, 57S17, 20F50, 32Q15, 57S15
Keywords: self-cover, holomorphic endomorphism, scale-invariant group, expanding map

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