The Baumslag--Solitar groups have presentations
$BS(p,q)=\langle a,b\mid a^p=b^{-1}a^q b\rangle$.
They have the following nice properties:
- they're two generator, one relator groups;
- they can be written as an HNN extension of $\mathbb{Z}$ over $\mathbb{Z}$. (This means that they're constructed by 'gluing' $\mathbb{Z}$ to itself in some way.)
So from the point of view of combinatorial group theory, they could hardly be simpler. And yet, for suitable values of $p$ and $q$ (typically $p,q$ relatively prime integers greater than 1 will do), we find that:
- they're non-Hopfian, meaning that they admit a self-epimorphism with non-trivial kernel;
- hence they're not even residually finite;
- they have exponential Dehn function (meaning that the word problem can be solved, but only very slowly);
- their virtual first Betti number is one (meaning that every finite-index subgroup has abelianisation of rank one)...
I could go on.
