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Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper $ 3$-realizability of groups


Authors: M. Cárdenas, F. F. Lasheras, A. Quintero and R. Roy
Journal: Proc. Amer. Math. Soc. 147 (2019), 1797-1804
MSC (2010): Primary 57M07; Secondary 57M10, 57M20
DOI: https://doi.org/10.1090/proc/14373
Published electronically: December 12, 2018
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Abstract: We recall that a finitely presented group $ G$ is properly $ 3$-realizable if for some finite $ 2$-dimensional CW-complex $ X$ with $ \pi _1(X) \cong G$, the universal cover $ \widetilde {X}$ has the proper homotopy type of a $ 3$-manifold. This purely topological property is closely related to the asymptotic behavior of the group $ G$. We show that proper $ 3$-realizability is also a geometric property meaning that it is a quasi-isometry invariant for finitely presented groups. In fact, in this paper we prove that (after taking wedge with a single $ n$-sphere) any two infinite quasi-isometric groups of type $ F_n$ ($ n \geq 2$) have universal covers whose $ n$-skeleta are proper homotopy equivalent. Recall that a group $ G$ is of type $ F_n$ if it admits a $ K(G,1)$-complex with finite $ n$-skeleton.


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Additional Information

M. Cárdenas
Affiliation: Departamento de Geometría y Topología, Fac. Matemáticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain
Email: mcard@us.es

F. F. Lasheras
Affiliation: Departamento de Geometría y Topología, Fac. Matemáticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain
Email: lasheras@us.es

A. Quintero
Affiliation: Departamento de Geometría y Topología, Fac. Matemáticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain
Email: quintero@us.es

R. Roy
Affiliation: College of Arts and Sciences, New York Institute of Technology, Old Westbury, New York 11568-8000
Email: rroy@nyit.edu

DOI: https://doi.org/10.1090/proc/14373
Keywords: Proper homotopy, quasi-isometry, properly $3$-realizable, $3$-manifold
Received by editor(s): April 10, 2017
Received by editor(s) in revised form: July 10, 2018
Published electronically: December 12, 2018
Additional Notes: This work was partially supported by the project MTM 2015-65397
Communicated by: Ken Bromberg
Article copyright: © Copyright 2018 American Mathematical Society