“Baumslag–Solitar group”的意思、由来-开放百科全书

词条

Baumslag–Solitar group

释义

Linear representation
See also
Notes
References

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 {{math|m}} and {{math|n}}, the Baumslag–Solitar group is denoted {{math|BS(m, n)}}. The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various {{math|BS(m, n)}} are well-known groups. {{math|BS(1, 1)}} is the free abelian group on two generators, and {{math|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

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

It is worth noting that this will not, in general, be an isomorphism. For instance if {{math|BS(m, n)}} is not residually finite (i.e. if it is not the case that {{math|{{!}}m{{!}} {{=}} 1}}, {{math|{{!}}n{{!}} {{=}} 1}}, or {{math|{{!}}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.^[2]

Notes

1. ^See [https://www.jstor.org/pss/1995962 Nonresidually Finite One-Relator Groups] by Stephen Meskin for a proof of the residual finiteness condition
2. ^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

References

{{springer|id=B/b130070|title=Baumslag–Solitar group|author=D.J. Collins}}
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}}

1 : Combinatorial group theory

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Linear representation

See also

Notes

References