M-Xylene (data page): Difference between revisions

Latest revision as of 23:33, 17 February 2013

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, b ∣ b a^{m} b^{- 1} = a^{n} ⟩ .

For each integer $m$ and $n$ , the Baumslag–Solitar group is denoted $B S (m, n)$ . The relation in the presentation is called the Baumslag–Solitar relation.

Some of the various $B S (m, n)$ are well-known groups. $B S (1, 1)$ is the free abelian group on two generators, and $B S (1, - 1)$ is the Klein bottle group.

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 $A = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ and $B = (\begin{matrix} \frac{n}{m} & 0 \\ 0 & 1 \end{matrix})$ . The matrix group $G$ generated by $A$ and $B$ is a homomorphic image of $B S (m, n)$ , via the homomorphism $a \mapsto A$ , $b \mapsto B$ .

It is worth noting that this will not, in general, be an isomorphism. For instance if $B S (m, n)$ 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 Mal'cev.^[2]

Notes

↑ See Nonresidually Finite One-Relator Groups by Stephen Meskin for a proof of the residual finiteness condition
↑ Anatoliĭ Ivanovich Mal'cev, "On the faithful representation of infinite groups by matrices" Transl. Amer. Math. Soc. (2), 45 (1965), pp. 1–18

References

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Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201. Template:MR

Template:Algebra-stub

[1] See 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" Transl. Amer. Math. Soc. (2), 45 (1965), pp. 1–18

[1]

[2]

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+[[File:Baumslag-Solitar Cayley.svg|thumb|350px|One sheet of the [[Cayley graph]] of the Baumslag–Solitar group <math>BS(1,2)</math>. Red edges correspond to <math>a</math> and blue edges correspond to <math>b</math>.]]
+[[File:Baumslag-Solitar Cayley 3D.svg|thumb|The sheets of the [[Cayley graph]] of the Baumslag-Solitar group <math>BS(1,2)</math> fit together into an infinite [[binary tree]].]]
+In the [[mathematics|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]]
+: <math>\langle a, b \mid b a^m b^{-1} = a^n \rangle.</math>
+For each integer <math>m</math> and <math>n</math>, the Baumslag–Solitar group is denoted <math>BS(m,n)</math>.  The relation in the presentation is called the '''Baumslag–Solitar relation'''.
+Some of the various <math>BS(m,n)</math> are well-known groups. <math>BS(1,1)</math> is the [[free abelian group]] on two [[Generating set of a group|generators]], and <math>BS(1,-1)</math> is the [[Klein bottle]] group.
+The groups were defined by [[Gilbert Baumslag]] and [[Donald Solitar]] in 1962 to provide examples of non-[[Hopfian group|Hopfian]] groups.  The groups contain [[residually finite]] groups, Hopfian groups that are not residually finite, and non-Hopfian groups.
+==Linear representation==
+Define <math>A=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)</math> and <math>B=\left(\begin{smallmatrix}\frac{n}{m}&0\\0&1\end{smallmatrix}\right)</math>. The matrix group <math>G</math> generated by <math>A</math> and <math>B</math> is a homomorphic image of  <math>BS(m,n)</math>, via the homomorphism <math>a\mapsto A</math>, <math>b\mapsto B</math>.
+It is worth noting that this will not, in general, be an isomorphism. For instance if <math>BS(m,n)</math> is not [[residually finite]] (i.e. if it is not the case that <math>|m|=1</math>, <math>|n|=1</math>, or <math>|m|=|n|</math><ref>See [http://www.jstor.org/pss/1995962 Nonresidually Finite One-Relator Groups] by Stephen Meskin for a proof of the residual finiteness condition</ref>) it cannot be isomorphic to a finitely generated [[linear group]], which is known to be [[residually finite]] by a theorem of Mal'cev.<ref>Anatoliĭ Ivanovich Mal'cev, "On the faithful representation of infinite groups by matrices" Transl. Amer. Math. Soc. (2), 45 (1965), pp. 1–18</ref>
+==Notes==
+<references/>
+==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}}
+{{DEFAULTSORT:Baumslag-Solitar group}}
+[[Category:Combinatorial group theory]]
+{{algebra-stub}}

M-Xylene (data page): Difference between revisions

Latest revision as of 23:33, 17 February 2013

Linear representation

Notes

References

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