Non-compact 3-manifolds proper homotopy equivalent to geometrically simply connected polyhedra and proper 3-realizability of groups Louis Funar 1 , Francisco F. Lasheras 2 and Duˇ san Repovˇ s 3 * 1 Institut Fourier BP 74, UFR Math´ ematiques, Univ.Grenoble I 38402 Saint-Martin-d’H` eres Cedex, France 2 Departamento de Geometria y Topologia, Universidad de Sevilla, Apdo 1160, 41080 Sevilla, Spain 3 Faculty of Mathematics and Physics, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia May 14, 2009 Abstract The principal result of this paper is a homotopy criterion for detecting the tameness of non-compact 3-manifolds which extends the one worked out by L.Funar and T.L.Thickstun for open 3-manifolds. A group is properly 3-realizable if it is the fundamental group of a compact 2-polyhedron whose universal covering is proper homotopy equivalent to a 3-manifold. As a consequence of the main result a properly 3-realizable group which is also quasi-simply filtered 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. AMS MOS Subj.Classification(1991): 57 M 50, 57 M 10, 57 M 30. Keywords and phrases: Properly 3-realizable, geometric simple connectivity, quasi-simple filtered group, missing boundary 3-manifold, Coxeter group. 1 Introduction In [18, 21] the authors proved that an open 3-manifold is simply connected at infinity if it has the proper homotopy type of a weakly geometrically simply connected polyhedron. The simple connectivity at infinity is a strong tameness condition for open 3-manifolds which, roughly speaking, expresses the fact that each end is collared by a 2-sphere. The main concern of this paper is to give a similar homotopy criterion for detecting the tameness in the case of 3-manifolds with boundary. The relevant tameness conditions have to be changed accordingly, in order to take into account the boundary behavior. Instead of the simple connectivity at infinity we will consider the so-called missing boundary manifold condition introduced by Simon (see [40]), while the weak geometric simple connectivity has to be replaced by the stronger pl-geometric simple connectivity, to be defined below. Despite the fact that there are similarities in the proofs with [18, 21], the case where manifolds have non-compact boundary components presents some new and interesting features. Working in this more general context opens the possibility to find applications to geometric group theory. Specifically, we obtain necessary conditions for a finitely presented group to act freely co-compactly on a simply connected 2-complex having the proper homotopy type of a 3-manifold. In particular we find explicit examples of groups which do not have this property. * Emails: funar@fourier.ujf-grenoble.fr (L.Funar), lasheras@us.es (F.F.Lasheras), dusan.repovs@guest.arnes.si (D.Repovˇ s) 1