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

For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various are well-known groups. is the free abelian group on two generators, and 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

The matrix group generated by and is a homomorphic image of , via the homomorphism induced by

This will not, in general, be an isomorphism. For instance if is not residually finite (i.e. if it is not the case that , , or See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.Anatoliĭ Ivanovich Mal'cev, "On the faithful representation of infinite groups by matrices" Translations of the American Mathematical Society (2), 45 (1965), pp. 1–18

History

The group first appeared in a 1951 paper of Graham Higman.Graham Higman, "A finitely related group with an isomorphic proper factor group" J.Lond.Math.Soc.26 (1951), pp. 59–61. It was for this reason that, according to Meier,John Meier, "Groups, Graphs and Trees", Cambridge University Press, (2008), page 100. "Baumslag [...] waged a vigorous, sustained, and ultimately doomed campaign against referring to as a Baumslag–Solitar group."

Baumslag–Solitar group

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