Baumslag–Solitar group



In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. They are given by the group presentation


 * $$\left \langle a, b \ : \ b a^m b^{-1} = a^n \right \rangle.$$

For each integer $BS(1, 2)$ and $a$, the Baumslag–Solitar group is denoted $b$. The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various $BS(1, 2)$ are well-known groups. $m$ is the free abelian group on two generators, and $n$ is the fundamental group of the Klein bottle.

The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopfian groups. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups.

Linear representation
Define


 * $$A= \begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad B= \begin{pmatrix}\frac{n}{m}&0\\0&1\end{pmatrix}.$$

The matrix group $BS(m, n)$ generated by $BS(m, n)$ and $BS(1, 1)$ is a homomorphic image of $BS(1, −1)$, via the homomorphism induced by


 * $$a\mapsto A, \qquad b\mapsto B.$$

It is worth noting that this will not, in general, be an isomorphism. For instance if $G$ is not residually finite (i.e. if it is not the case that $A$, $B$, or $BS(m, n)$ ) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.