On simplicial resolutions of groups

The Problems of Dimension Four, and Some Ramifications

“…where j is an inclusion map, and f is some simplicial map satisfying the Dehn-type property j(k) ∩ M 2 ( f ) = ∅ (where M 2 ( f ) ⊂ K denotes the set of double points of f ). This notion was introduced by Stephen Brick and Mike Mihalik in [40] (but see also [28,29,41]), and they also proved the following useful concepts:…”

“…It concerns finitely presented groups (no other ones will be considered here) and the geometric and topological properties of the universal covering spaces of the compact spaces having as fundamental group, the group in question. According to the viewpoint of the quasi-isometries of Misha Gromov [27], the groups and those universal covering spaces are in fact equivalent objects, and so many topological properties make sense for groups as well (see e.g., [28,29]). Now, when Grisha Perelman proved the Poincaré Conjecture and Thurston's geometrization, this had very important consequences for group theory as well.…”

On the Geometry and Topology of Discrete Groups: An Overview

“…Another interesting implication of the simple connectivity at infinity comes from its connection with the so-called Universal Covering Conjecture. Since the 1960s, topologists have studied the behavior at infinity of contractible universal covering spaces of closed 3manifolds and proposed the following problem/conjecture (for a more historical panoramic view, see [9,19]):…”

“…The main tool for proving the last theorems of the previous section was the following notion, invented and developed by Poénaru in [27] and thereafter utilized in his scientific work (see [28] but also [19,26,29]): Definition 12. Let M 3 be a 3-manifold.…”