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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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 PDF
Proc. Amer. Math. Soc. 147 (2019), 1797-1804 Request permission

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
  • 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