#### Volume 22, issue 4 (2018)

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Towers of regular self-covers and linear endomorphisms of tori

### Wouter van Limbeek

Geometry & Topology 22 (2018) 2427–2464
##### 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.

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##### Keywords
self-cover, holomorphic endomorphism, scale-invariant group, expanding map
##### Mathematical Subject Classification 2010
Primary: 57N99, 57S17
Secondary: 20F50, 32Q15, 57S15
##### Publication
Received: 12 December 2016
Revised: 22 July 2017
Accepted: 2 September 2017
Published: 5 April 2018
Proposed: Walter Neumann
Seconded: Benson Farb, Leonid Polterovich
##### Authors
 Wouter van Limbeek Department of Mathematics University of Michigan Ann Arbor, MI United States