Article
Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper 3-realizability of groups
Author/s | Cardenas Escudero, Manuel
Fernández Lasheras, Francisco Jesús Quintero Toscano, Antonio Rafael Roy, R. |
Department | Universidad de Sevilla. Departamento de Geometría y Topología |
Publication Date | 2019-04-01 |
Deposit Date | 2023-12-18 |
Published in |
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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 π1(X) ∼= G, the universal
cover X has the proper homotopy type of a 3-manifold. This purely ... We recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with π1(X) ∼= G, the universal cover 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 quasiisometric groups of type Fn (n ≥ 2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group G is of type Fn if it admits a K(G, 1)-complex with finite n-skeleton. |
Citation | Cardenas Escudero, M., Fernández Lasheras, F.J., Quintero Toscano, A.R. y Roy, R. (2019). Quasi-isometries and proper homotopy: The quasi-isometry invariance of proper 3-realizability of groups. Proceeding of the American Mathematical Society, 147 (4), 1797-1804. https://doi.org/10.1090/proc/14373. |
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