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

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

doi:10.1090/proc/14373

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.

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