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Towers of regular self-covers and linear endomorphisms of tori

Wouter van Limbeek

Geometry & Topology 22 (2018) 2427–2464

Let M be a closed manifold that admits a self-cover p: M M of degree > 1. We say p is strongly regular if all iterates pn: M M are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of M: We prove that π1(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.

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