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QUASI-ISOMETRIES AND PROPER HOMOTOPY: THE QUASI-ISOMETRY INVARIANCE OF PROPER 3-REALIZABILITY OF GROUPS.

Authors :
ĆARDENAS, M.
LASHERAS, F. F.
QUINTERO, A.
ROY, R.
Source :
Proceedings of the American Mathematical Society; Apr2019, Vol. 147 Issue 4, p1797-1804, 8p
Publication Year :
2019

Abstract

We recall that a finitely presented group G is properly 3-realizable if for some finite 2-dimensional CW-complex X with π<subscript>1</subscript>(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 F<subscript>n</subscript> (n ≥ 2) have universal covers whose n-skeleta are proper homotopy equivalent. Recall that a group G is of type F<subscript>n</subscript> if it admits a K(G, 1)-complex with finite n-skeleton. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00029939
Volume :
147
Issue :
4
Database :
Complementary Index
Journal :
Proceedings of the American Mathematical Society
Publication Type :
Academic Journal
Accession number :
134647362
Full Text :
https://doi.org/10.1090/proc/14373