Baumslag–Solitar group

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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

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

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

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

Since a theorem of Malc'ev states that a finitely generated group which admits a faithful finite-dimensional linear representation over the field of complex numbers (in short : a finitely generated linear group) is residually finite (and hence Hopfian), some Baumslag-Solitar groups (for instance B(2,3)) are examples of groups without faithful finite-dimensional representations.