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^{-1} = a^n angle.

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.

Linear representation

Define A=ig(egin{smallmatrix}1&1\0&1end{smallmatrix}ig) and B=ig(egin{smallmatrix}frac{n}{m}&0\0&1end{smallmatrix}ig). The matrix group G generated by A and B is isomorphic to B(m,n), 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), 199–201. MathSciNet|id=0142635


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