Baumslag–Solitar group

One sheet of the Cayley graph of the Baumslag–Solitar group Template:Math. Red edges correspond to Template:Math and blue edges correspond to Template:Math.

The sheets of the Cayley graph of the Baumslag-Solitar group Template:Math fit together into an infinite binary tree.

Visualization comparing the sheet and the binary tree Cayley graph of $\text{BS}(1,2)$. Red and blue edges correspond to $a$ and $b$, respectively.

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

$⟨a,b:b{a}^m{b}^{-1}=a^n⟩.$

For each integer Template:Math and Template:Math, the Baumslag–Solitar group is denoted Template:Math. The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various Template:Math are well-known groups. Template:Math is the free abelian group on two generators, and Template:Math 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.

Linear representation

Define

$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),B=\left(\begin{array}{cc}\frac{n}{m}& 0\\ 0& 1\end{array}\right).$

The matrix group Template:Math generated by Template:Math and Template:Math is a homomorphic image of Template:Math, via the homomorphism induced by

$a\mapsto A,b\mapsto B.$

This will not, in general, be an isomorphism. For instance if Template:Math is not residually finite (i.e. if it is not the case that Template:Math, Template:Math, or Template:Math) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.

See also

• Binary tiling
• Solv geometry

References

• Template:Springer
• Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. Template:MR