On accessible groups

Decomposition theorems for group pairs

“…In Case 3), A a may not fulfill property (i); but using the result of Swarup [16] and the fact that A1 is finitely generated and accessible, we can find a factor (in the sense of [1], p. 341) of A~, say B~, which contains Gz and fulfills property (i) and still the properties (ii) and (iii). Since B~ is a factor of A, we know that B 1 <A (see [1] and [7] We say in the following that such a set has form *).…”

Accessibility and Groups of Cohomological Dimension One

“…Recall that D(G,H) = Kerp H where p H : T>er{G,ZG) -* Ver(H,ZG) is the restriction mapping. We confine our attention now to the case where R = Z, and generalize the results of [1]. Let X> 1 ((?, H) be the submodule of Dei(G,ZG) generated by D{G,H) and e G .…”

“…HH is finitely generated with at most one end-in particular if H is finitethen H^H^ZG)^®, by Lemma 5.3, and so A(G,H) = A{G), in the notation of [1]. If S is a subgroup of G for which (S,SnH) is finitely generated, then as in [1, p. 337], there is induced a homomorphism…”

“…; e K as a product of elements in (HnK)uSuS- 1 , we can find keK and s e Su S -1 with uh e B, uhs e B*. Then uk lies in one of the finitely many sets BnB^s' 1 . Thus there are only finitely many choices for uk, that is, only finitely many cosets gK meeting both B and JB*.…”

Plane motion groups and virtual Poincar� duality of dimension two

“…A group G is said to be of type (FP,,) if there is a projective 7lG-resolution of Z which is finitely generated in dimensions < n. The group G is said to split over a subgroup S if it is either an amalgamated free product G=G 1 *sG2, G 1 4=S+G2, or an HNN-extension G =G 1 *S,p. For the concept of"accessibility" of a group see, for example, [1].…”