On 3-dimensional wgsc inverse-representations of groups

Topics in Geometric Group Theory. Part I

“…The interest of the concept easy is that any easy group G is actually qsf [39] (as we shall see in the sequel of this paper by the second author), and this also means that if by any chance G = π 1 M 3 , then π ∞ 1 M 3 = 0 [44]. We will come back to these issues later.…”

“…First of all in [43] it is proved that almost-convex groups are tame 1-combable (see the next section for definitions and more details) and hence qsf by [32]. By the main result of [44], one can infer that such a group admits an easy wgsc-representation with an additional finiteness condition. This is weaker than being an easy group (where it is required an easy gsc-representation).…”

“…Remark 3.2. Another proof of a slightly weaker form of Proposition 1 can be deduced from the recent papers [43,44]. Although this proof is very short, it relies on other deep results.…”

“…An old classical easy fact specific for dimension three is the following one: an open 3-manifold V 3 which is wgsc is also simply connected at infinity (see e.g. [44]). In higher dimension (n ≥ 5), it turns out that open simply connected n-manifolds which are also simply connected at infinity are actually gsc [64].…”

“…This is weaker than being an easy group (where it is required an easy gsc-representation). However, in dimension 3, this condition actually implies the simple connectivity at infinity (see [44]), namely the original motivation of [51].…”