2016/09/21 by Wouter van Limbeek, Wouter Van Limbeek · 4 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic number #Cover (algebra) #Endomorphism #Geometric and Algebraic Topology #Geometry and complex manifolds #Holomorphic function #Iterated function #Manifold (fluid mechanics) #Torus #math.DG #math.DS #math.GT
paper · pdf · doi:10.2140/gt.2018.22.2427
published in Geometry & Topology 22(4), 2427-2464 (Mathematical Sciences Publishers) · 28 pages
arxiv created 2016/09/21 · openalex created_date 2016/09/30 · openalex publication_date 2018/04/05 · arxiv updated 2018/04/18 · openalex updated_date 2026/08/05
Let [math] be a closed manifold that admits a self-cover [math] of degree [math] . We say [math] is strongly regular if all iterates [math] are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of [math] : We prove that [math] surjects onto a nontrivial free abelian group [math] , and the self-cover is induced by a linear endomorphism of [math] . Under further hypotheses we show that a finite cover of [math] admits the structure of a principal torus bundle. We show that this applies when [math] is Kähler and [math] is a strongly regular, holomorphic self-cover, and prove that a finite cover of [math] splits as a product with a torus factor.