# 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

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

The matrix group *G* generated by *A* and *B* is a homomorphic image of BS(*m*, *n*), via the homomorphism induced by

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*|^{[1]}) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Mal'cev.^{[2]}

## Notes

- ↑ See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition
- ↑ Anatoliĭ Ivanovich Mal'cev, "On the faithful representation of infinite groups by matrices" Transl. Amer. Math. Soc. (2), 45 (1965), pp. 1–18

## 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