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One-relator groups and proper 3-realizability


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Opened Access One-relator groups and proper 3-realizability
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Author: Cárdenas Escudero, Manuel Enrique
Fernández Lasheras, Francisco Jesús
Quintero Toscano, Antonio Rafael
Repovš, Dušan
Department: Universidad de Sevilla. Departamento de Geometría y Topología
Date: 2009
Published in: Revista Matemática Iberoamericana, 25, 739-756.
Document type: Article
Abstract: How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π1(K) ∼= G whose universal cover K˜ has the proper homotopy type of a PL 3-manifold (with boundary). In this paper, we study the asymptotic behavior of finitely generated one-relator groups and show that those having finitely many ends are properly 3-realizable, by describing what the fundamental pro-group looks like, showing a property of one-relator groups which is stronger than the QSF property of Brick (from the proper homotopy viewpoint) and giving an alternative proof of the fact that one-relator groups are semistable at infinity.
Cite: Cárdenas Escudero, M.E., Fernández Lasheras, F.J., Quintero Toscano, A.R. y Repovš, D. (2009). One-relator groups and proper 3-realizability. Revista Matemática Iberoamericana, 25, 739-756.
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