Exercise 29, § 7

Let $G = G_1 *_{A}G_2$ be the sum of two groups $G_1$ and $G_2$ amalgamated by a subgroup $A$. a) Now prove that $G$ is of finite type if $G_1$ and $G_2$ are of finite type. b) Suppose that $G_1$ and $G_2$ are of finite presentation. Now prove the equivalence of the following two properties: (1) $G$ is of finite presentation. (2) $A$ is of finite type. (One will first prove that (1) implies (2). On the other hand, if $A$ is not of finite type, there exists an increasing filtered family $(A_i)_{i \in I}$ of subgroups of $A$, with $\bigcup_{i \in I} A_i = A$ and $A_i \neq A$ for every $i$. Let $H_i$ (resp. $H$) be the kernel of the canonical homomorphism of $G_1 *_{A_i} G_2$ onto $G_1 *_{A_i} G_2$ (resp. onto $G$). One will prove that $H$ is the union of the $H_i$ and that $H_i \neq H$ for every $i$; apply then exerc. 16 d) of I, p. 145.) c) Deduce an example of a group of finite type which is not of finite presentation. (Take for $G_1$ and $G_2$ free groups of rank 2 and for $A$ a free group of infinite rank, cf. I, p. 147, exerc. 22.)