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.

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.

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

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 .
Animated depiction of the relation between the "sheet" and the full infinite binary tree Cayley graph of BS(1,2)
Visualization comparing the sheet and the binary tree Cayley graph of . Red and blue edges correspond to and , respectively.