Baumslag–Solitar group

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.

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

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

a\mapsto A,\qquad 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 (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 Mal'cev.^[2]

Notes

↑ See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition
↑ Anatoliĭ Ivanovich 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) [1994], "Baumslag–Solitar group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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

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