Baumslag–Solitar group

File:Baumslag-Solitar Cayley.svg

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.

File:Baumslag-Solitar Cayley 3D.svg

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

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

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 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 G generated by A and B is a homomorphic image of BS(m, n), 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 BS(m, n) is not residually finite (i.e. if it is not the case that |m| = 1, |n| = 1, or |m| = |n|) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Mal'cev.

References

• Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. MR 0142635