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One sheet of the Cayley graph of the Baumslag–Solitar group $BS(1,2)$. Red edges correspond to $a$ and blue edges correspond to $b$.

The sheets of the Cayley graph of the Baumslag-Solitar group $BS(1,2)$ fit together into an infinite binary tree.

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

$⟨a,b\mid b{a}^m{b}^{-1}=a^n⟩.$

For each integer $m$ and $n$, the Baumslag–Solitar group is denoted $BS(m,n)$. The relation in the presentation is called the Baumslag–Solitar relation.

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

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 $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ and $B=\left(\begin{array}{cc}\frac{n}{m}& 0\\ 0& 1\end{array}\right)$. The matrix group $G$ generated by $A$ and $B$ is a homomorphic image of $BS(m,n)$, via the homomorphism $a\mapsto A$, $b\mapsto B$.

It is worth noting that this will not, in general, be an isomorphism. For instance if $BS(m,n)$ is not residually finite (i.e. if it is not the case that $\left|m\right|=1$, $\left|n\right|=1$, or $\left|m\right|=\left|n\right|$) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Mal'cev.

References

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• Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. Template:MR

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