Properly 3-realizable groups

Ayala, R.; Cárdenas, M.; Lasheras, F.; Quintero, A.

doi:10.1090/S0002-9939-04-07628-2

Properly $3$-realizable groups
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by R. Ayala, M. Cárdenas, F. F. Lasheras and A. Quintero

Proc. Amer. Math. Soc. 133 (2005), 1527-1535

DOI: https://doi.org/10.1090/S0002-9939-04-07628-2

Published electronically: November 19, 2004

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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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