Towers of regular self-covers and linear endomorphisms of tori

van Limbeek, Wouter

doi:10.2140/gt.2018.22.2427

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Towers of regular self-covers and linear endomorphisms of tori

Wouter van Limbeek

Geometry & Topology 22 (2018) 2427–2464

DOI: 10.2140/gt.2018.22.2427

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 $π_{1} (M)$ 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 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.

Keywords

self-cover, holomorphic endomorphism, scale-invariant group, expanding map

Mathematical Subject Classification 2010

Primary: 57N99, 57S17

Secondary: 20F50, 32Q15, 57S15

References

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

Volume 22, issue 4 (2018)