Constructable solvable groups

Virtually soluble groups of type $\operatorname{FP}_∞$

“…Most of the remainder of this paper is then devoted to proving that centralizers of finite subgroups are also of type FP 1 . These centralizers are finitely presented [1] and have finite index in the corresponding normalizers hence the conditions of Theorem 1.2 are satisfied. Furthermore, since a virtually soluble group of type FP 1 also admits a finite dimensional model for EG, the claim that it admits a finitely dominated model for EG now follows from Theorems 5.1 and 6.3 in [22].…”

“…Groups satisfying the conditions of the theorem are minimax, i.e. have a finite series of normal subgroups such that each factor has either min or max, see for example [2, 7.16] and are finitely presented [1]. Finite presentability of soluble groups of type FP 1 implies in particular that these are of type VF: groups of type VF are those having a finite-index subgroup admitting a finite K.G; 1/.…”

“…Proof. As remarked in the introduction, constructible groups are finitely presented (see [1]) so also G and all its Weyl groups are. All that remains to prove is that for any finite subgroup F of G, the centralizer C G .F / and therefore the normalizer N G .F / is of type FP 1 .…”

Soluble groups with a finite rewriting system

“…Note that, as i i e l c^G ) , the notation <j>~x(v) is well-defined. Let R" = R'uJ?,uR 2 uK 3 . We claim that (L",R") is a finite complete rewriting system for H. Firstly, note that the /?…”

“…So G is constructible if it is n-constructible for some n. In the case of soluble groups, this simplifies considerably. It follows from the work in [3] that a soluble group is constructible precisely when it is a finite extension of a torsion-free group which can be obtained by a finite series of HNN-extensions in each of which one of the associated subgroups equals the base group and the other has finite index in the base group.…”