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

${\displaystyle \left\langle a,b\$ :\ ba^{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.