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

${\displaystyle \left\langle a,b\$ :\ ba^{m}b^{-1}=a^{n}\right\rangle .}

One sheet of the Cayley graph of the Baumslag–Solitar group BS(1, 2). Red edges correspond to a and blue edges correspond to b.

The sheets of the Cayley graph of the Baumslag-Solitar group BS(1, 2) fit together into an infinite binary tree.

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

For each integer m and n, the Baumslag–Solitar group is denoted BS(m, n). The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various BS(m, n) are well-known groups. BS(1, 1) is the free abelian group on two generators, and BS(1, −1) 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

${\displaystyle A={\begin{pmatrix}1&1\\0&1\end{pmatrix}},\qquad B={\begin{pmatrix}{\frac {n}{m}}&0\\0&1\end{pmatrix}}.}$

The matrix group G generated by A and B is a homomorphic image of BS(m, n), via the homomorphism induced by

${\displaystyle a\mapsto A,\qquad b\mapsto B.}$

This will not, in general, be an isomorphism. For instance if BS(m, n) is not residually finite (i.e. if it is not the case that |m| = 1, |n| = 1, or |m| = |n|^[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 BS(1, 2) 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 BS(1, 2) as a Baumslag–Solitar group."

See also

• Binary tiling
• Solv geometry