How different is the universal cover of a given finite $2$-complex
from a $3$-manifold (from the proper homotopy viewpoint)?
Regarding this question, we recall that a finitely presented group
$G$ is said to be properly $3$-realizable if there exists a
compact $2$-polyhedron $K$ with $\pi_1(K) \cong G$ whose universal
cover $\tilde{K}$ has the proper homotopy type of a PL
$3$-manifold (with boundary). In this paper, we study the
asymptotic behavior of finitely generated one-relator groups and
show that those having finitely many ends are properly
$3$-realizable, by describing what the fundamental pro-group looks
like, showing a property of one-relator groups which is stronger
than the QSF property of Brick (from the proper homotopy
viewpoint) and giving an alternative proof of the fact that
one-relator groups are semistable at infinity.
Publié le : 2009-06-15
Classification:
proper homotopy equivalence,
polyhedron,
one-relator group,
proper $3$-realizability,
end of group,
57M07,
57M10,
57M20
@article{1255440073,
author = {C\'ardenas
,
Manuel and Lasheras
,
Francisco F. and Quintero
,
Antonio and Repov\v s
,
Du\v san},
title = {One-relator groups and proper $3$-realizability},
journal = {Rev. Mat. Iberoamericana},
volume = {25},
number = {1},
year = {2009},
pages = { 739-756},
language = {en},
url = {http://dml.mathdoc.fr/item/1255440073}
}
Cárdenas
,
Manuel; Lasheras
,
Francisco F.; Quintero
,
Antonio; Repovš
,
Dušan. One-relator groups and proper $3$-realizability. Rev. Mat. Iberoamericana, Tome 25 (2009) no. 1, pp. 739-756. http://gdmltest.u-ga.fr/item/1255440073/