Английская Википедия:Baumslag–Solitar group

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Файл:Baumslag-Solitar Cayley.svg

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

Файл:Baumslag-Solitar Cayley 3D.svg

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

Animated depiction of the relation between the "sheet" and the full infinite binary tree Cayley graph of BS(1,2)

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

<math>\left \langle a, b \ : \ b a^m b^{-1} = a^n \right \rangle.</math>

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

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

<math>A= \begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad B= \begin{pmatrix}\frac{n}{m}&0\\0&1\end{pmatrix}.</math>

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

<math>a\mapsto A, \qquad b\mapsto B.</math>

It is worth noting that this will not, in general, be an isomorphism. For instance if Шаблон:Math is not residually finite (i.e. if it is not the case that Шаблон:Math, Шаблон:Math, or Шаблон: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

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