Examples of non-finitely presented groups

Question

I know several constructions leading to finitely generated non-finitely presented groups, using amalgamated products:

Property: Let $A,B$ be two finitely presented groups. Then $A \underset{C}{\ast} B$ is finitely presented iff $C$ is finitely generated.

using HNN extensions:

Property: Let $A$ be a finitely presented group. Then $\underset{C}{\ast} A$ is finitely presented iff $C$ is finitely generated.

or using wreath products (more difficult result):

Property: Let $A,B$ be two finitely presented groups. Then $A \wr B$ is finitely presented iff $A$ is trivial or $B$ is finite.

However, the only application that I know giving a "nice" group, that is a group with a simple description (not using a presentation of course), is the lamplighter group $L_2= \mathbb{Z}_2 \wr \mathbb{Z}$. Do you know other examples?

Answer 1

For $n \ge 1$, consider the homomorphism of a direct product $F_2^n$ of $n$ copies of the free group of rank $2$ to ${\mathbb Z}$ in which all generators aret mapped to $1$, and let $K_n$ be its kernel.

Then $K_1$ is not finitely generated, and $K_2$ is finitely generated but not finitely presented. They are finitely presented for $n \ge 3$, but I believe that they satisfy interesting homological conditions: I don't remember the details right now!