# Baumslag–Solitar group

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

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

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

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

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

Linear representation

Define and . The matrix group $G$ generated by $A$ and $B$ is isomorphic to $B\left(m,n\right)$, via the isomorphism $Amapsto a$, $Bmapsto b$.

References

*
* Gilbert Baumslag and Donald Solitar, [http://projecteuclid.org/euclid.bams/1183524561 "Some two-generator one-relator non-Hopfian groups"] , Bulletin of the American Mathematical Society 68 (1962), 199&ndash;201. MathSciNet|id=0142635

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