M-Xylene (data page): Difference between revisions

Latest revision as of 00:33, 18 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

\langle a,b\mid ba^{m}b^{-1}=a^{n}\rangle .

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

Some of the various $BS(m,n)$ are well-known groups. $BS(1,1)$ is the free abelian group on two generators, and $BS(1,-1)$ 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 $A=\left({\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}\right)$ and $B=\left({\begin{smallmatrix}{\frac {n}{m}}&0\\0&1\end{smallmatrix}}\right)$ . The matrix group $G$ generated by $A$ and $B$ is a homomorphic image of $BS(m,n)$ , 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 $BS(m,n)$ 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 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

Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

my web-site http://himerka.com/
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]

@@ Line 1: / Line 1: @@
-Hi there, I am Yoshiko Villareal but I by no means really favored that title. Climbing is what love doing. Alabama has always been his house. Bookkeeping is what he does.<br><br>Stop by my web-site extended car warranty ([http://Www.gettingtherefromhere.info/User-Profile/userId/13851 click through the up coming web site])
+[[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 00:33, 18 February 2013

Linear representation

Notes

References

Navigation menu

Search