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
A = ( 1 1 0 1 ) , B = ( n m 0 0 1 ) . {\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 a ↦ A , b ↦ B . {\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|) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.
Imagem: Jim.belk · CC0 · Openverse
The group BS(1, 2) first appeared in a 1951 paper of Graham Higman. It was for this reason that, according to Meier, "Baumslag [...] waged a vigorous, sustained, and ultimately doomed campaign against referring to BS(1, 2) as a Baumslag–Solitar group."
