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.
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Louis Funar, Francisco F. Lasheras, Dušan D. Repovš, Groups which are not properly 3-realizable. Rev. Mat. Iberoam. 28 (2012), no. 2, pp. 401–414