Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper $3$-realizability of groups
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- by M. Cárdenas, F. F. Lasheras, A. Quintero and R. Roy
- Proc. Amer. Math. Soc. 147 (2019), 1797-1804
- 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.References
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Bibliographic 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
- MR Author ID: 633766
- Email: lasheras@us.es
- A. Quintero
- Affiliation: Departamento de Geometría y Topología, Fac. Matemáticas, Universidad de Sevilla, C/. Tarfia s/n 41012-Sevilla, Spain
- MR Author ID: 143190
- 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
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1797-1804
- MSC (2010): Primary 57M07; Secondary 57M10, 57M20
- DOI: https://doi.org/10.1090/proc/14373
- MathSciNet review: 3910444