The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup

Abstract

Abstract.We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G.

Spin covers of maximal compact subgroups of Kac–Moody groups and spin-extended Weyl groups

“…Moreover, U is finite. Therefore by [29,Theorem 6], each˛.K i U / is conjugate to K 1 U or to K 2 U . Hence˛.K 1 U /,˛.K 2 U / each stabilise a vertex of the Bruhat-Tits tree X of G. Sincę…”

Symmetries and reversing symmetries of polynomial automorphisms of the plane

“…It was later put into a more general framework in [18], and a complete account is also contained in [37, Sec. 0].…”

“…Proof. First, consider the affine involutions which must satisfy (18). From Lemma 4, one possibility based around M = I is g ∈ B defined by x ′ = −x + u, y ′ = −y + v, with arbitrary u, v ∈ K. Taking h ∈ B via x ′ = x − u/2 and y ′ = y − v/2, one finds hgh −1 is the linear map defined by I.…”

Finite and infinite cyclic extensions of free groups

“…Karrass, A. Pietrowski and D. Solitar [5] ,K; rel K, tFt~l = where F is finite and x(K) is defined. If we apply these formulas to the given group G, we obtain an expression for X(G) = (1 -r) U in terms of the characteristics of the component groups, and the generalized Schreier rank formula follows immediately.…”