Baumslag–Solitar group

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Template:Short description Template:Dark mode invert Template:Dark mode invert Template:Dark mode invert 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 : bamb−1=an⟩.

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=(1101),B=(nm001).

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↦A,b↦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[1]) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.[2]

History

The group Template:Math first appeared in a 1951 paper of Graham Higman.[3] It was for this reason that, according to Meier,[4] "Baumslag [...] waged a vigorous, sustained, and ultimately doomed campaign against referring to Template:Math as a Baumslag–Solitar group."

See also

Notes

  1. ↑ See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition
  2. ↑ Anatoliĭ Ivanovich Mal'cev, "On the faithful representation of infinite groups by matrices" Translations of the American Mathematical Society (2), 45 (1965), pp. 1–18
  3. ↑ Graham Higman, "A finitely related group with an isomorphic proper factor group" J.Lond.Math.Soc.26 (1951), pp. 59–61.
  4. ↑ John Meier, "Groups, Graphs and Trees", Cambridge University Press, (2008), page 100.

References

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