A finitely presented HNN-extension example: Non-finitely generated groups, and an application of small cancellation theory

Decompositions of finitely generated and finitely presented groups

In this paper we discuss the splitting or decomposing of finitely generated groups into free products, free products with amalgamation or HNN extensions and we discuss the JSJ decomposition of finitely presented groups.

Finitely generated nilpotent groups are finitely presented and residually finite

Definition 1. Let G be a group. G is said to be residually finite if the intersection of all normal subgroups of G of finite index in G is trivial. For a survey of results on residual finiteness and related properties, see Mag-nus, Karrass, and Solitar [6, Section 6.5]. We shall present a proof of the following well known theorem, which is important for Kharlampovich [4, 5]. See also O. V. Bele...

Finitely Generated Abelian Groups

Finitely Generated Semiautomatic Groups

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.

Finitely Presented Residually Free Groups

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all n ∈ N, a residually free group is of type FPn if and only if it is of type Fn. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups ...