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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 \ : \ b a^m b^ = a^n \right \rangle. 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 In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
on two generators, and is the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
. The groups were defined by Gilbert Baumslag and
Donald Solitar Donald Solitar (September 5, 1932 in Brooklyn, New York, United States – April 28, 2008 in Toronto, Canada) was an American and Canadian mathematician, known for his work in combinatorial group theory.. Reprinted as and as . The Baumslag–Solit ...
in 1962 to provide examples of non-
Hopfian group In mathematics, a Hopfian group is a group ''G'' for which every epimorphism :''G'' → ''G'' is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group ''G'' is co-Hopfian if ...
s. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups.


Linear representation

Define :A= \begin1&1\\0&1\end, \qquad B= \begin\frac&0\\0&1\end. The matrix group generated by and is a homomorphic image of , 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 is not residually finite (i.e. if it is not the case that , , or ) 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


See also

* Binary tiling


Notes


References

* * Gilbert Baumslag and Donald Solitar, ''Some two-generator one-relator non-Hopfian groups'', Bulletin of the American Mathematical Society 68 (1962), 199–201. {{DEFAULTSORT:Baumslag-Solitar group Combinatorial group theory