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

$\langle a, b \mid b a^m b^{-1} = a^n \rangle.$

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 Klein bottle group.

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 $A=\big(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\big)$ and $B=\big(\begin{smallmatrix}\frac{n}{m}&0\\0&1\end{smallmatrix}\big)$ . The matrix group $G$ generated by $A$ and $B$ is a homomorphic image of $BS(m,n)$ , via the homomorphism $a\mapsto A$ , $b\mapsto B$ .

It is worth noting that this will not, in general, be an isomorphism. For instance if $BS(m,n)$ is not residually finite (ie, 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 Mal'cev^[2].

Notes

^ See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition
^ A. I. Mal'cev, "On the faithful representation of infinite groups by matrices" Transl. Amer. Math. Soc. (2) , 45 (1965) pp. 1–18

References

D.J. Collins (2001), "Baumslag–Solitar group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=B/b130070
Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. MR 0142635