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Properly $3$-realizable groups


Authors: R. Ayala, M. Cárdenas, F. F. Lasheras and A. Quintero
Journal: Proc. Amer. Math. Soc. 133 (2005), 1527-1535
MSC (2000): Primary 57M07; Secondary 57M10, 57M20
DOI: https://doi.org/10.1090/S0002-9939-04-07628-2
Published electronically: November 19, 2004
MathSciNet review: 2111954
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Abstract: A finitely presented group $G$ is said to be properly $3$-realizable if there exists a compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$ and whose universal cover $\tilde{K}$ has the proper homotopy type of a (p.l.) $3$-manifold with boundary. In this paper we show that, after taking wedge with a $2$-sphere, this property does not depend on the choice of the compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$. We also show that (i) all $0$-ended and $2$-ended groups are properly $3$-realizable, and (ii) the class of properly $3$-realizable groups is closed under amalgamated free products (HNN-extensions) over a finite cyclic group (as a step towards proving that $\infty$-ended groups are properly $3$-realizable, assuming $1$-ended groups are).


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Additional Information

R. Ayala
Affiliation: Departamento de Geometría y Topología, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, Spain

M. Cárdenas
Affiliation: Departamento de Geometría y Topología, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, Spain

F. F. Lasheras
Affiliation: Departamento de Geometría y Topología, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, Spain
Email: lasheras@us.es

A. Quintero
Affiliation: Departamento de Geometría y Topología, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, Spain

DOI: https://doi.org/10.1090/S0002-9939-04-07628-2
Received by editor(s): September 29, 2003
Received by editor(s) in revised form: December 31, 2003
Published electronically: November 19, 2004
Additional Notes: This work was partially supported by the project BFM 2001-3195-C02
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2004 American Mathematical Society

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