Revista Matemática Iberoamericana
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Groups which are not properly 3-realizable
Louis Funar (1), Francisco F. Lasheras (2) and Dušan Repovš (3)(1) Institut Fourier, UMR 5582, Université Grenoble I, 100 rue des Maths, B.P. 74, 38402, SAINT MARTIN D'HERES CEDEX, FRANCE
(2) Dpto. de Geometría y Topología, Universidad de Sevilla, Apartado postal 1160, 41080, SEVILLA, SPAIN
(3) Institute of Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, 1001, LJUBLJANA, SLOVENIA
A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provide the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.
Keywords: Properly 3-realizable, geometric simple connectivity, quasi-simple filtered group, Coxeter group