- 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 theoryand geometric group theoryas (counter)examples and test-cases. They are given by the group presentation
For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation.
Some of the various are well-known groups. is the
free abelian groupon two generators, and is the Klein bottlegroup.
These groups were defined by
Gilbert Baumslagand Donald Solitarin 1962 to provide examples of non-Hopfian groups. The class of Baumslag–Solitar groups contains residually finitegroups, Hopfian groups that are not residually finite, and non-Hopfian groups.
Define and . The matrix group generated by and is isomorphic to , via the isomorphism , .
* Gilbert Baumslag and Donald Solitar, [http://projecteuclid.org/euclid.bams/1183524561 "Some two-generator one-relator non-Hopfian groups"] ,
Bulletin of the American Mathematical Society68 (1962), 199–201. MathSciNet|id=0142635
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