By Katalin A. Bencsath, Marianna C. Bonanome, Margaret H. Dean, Marcos Zyman
Foreword.- Preface.- Preliminaries.- instruments: shows and their Calculus.- Constructions.- Representations and a Theorem of Krasner and Kaloujnine.- The Bieri-Strebel Theorems.- Finitely Generated Metabelian Groups.- An Embedding Theorem for Finitely Generated Metabelian Groups.- cartoon of facts of Lemma 1.1.- Theorem 2.1 Details.- offering an (Internal) HNN-Extension.- References
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Extra resources for Lectures on Finitely Generated Solvable Groups
Example text
Acta Sci. Mz.
Bonanome, Margaret H. 1 (Bieri and Strebel). Let N be a normal subgroup of a finitely presented group G. If G/N is infinite cyclic, then G is an ascending HNN-extension of a finitely generated group or contains a free subgroup of rank 2. Proof. We will show that any finitely presented group that has the infinite cyclic group as its homomorphic image is an HNN-extension of a finitely generated group. Since G/N is infinite cyclic, we can choose t ∈ G of infinite order, such that G/N = gp(tN). In particular, this implies that G = gp (t, N).
Thus, W has an infinite cyclic quotient. 1, there exists a finitely generated group A such that A ≤ B and either −1 At = B or At = B. In either case, B would be finitely generated, a contradiction. Hence, W is not finitely presented. To produce a finitely presented ascending HNN-extension of W , consider the subgroup of W generated by aat and t. Put b = b0 = aat = aa1 , b1 = bt0 = a1 a2 , . . , bn = btn−1 = an an+1 , . . It is easy to verify that the subgroup of W generated by {. . , b−2 , b−1 , b0 , b1 , b2 , .