Difference between revisions of "Special linear group:SL(2,Z)"

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* The [[defining ingredient::inner automorphism group]] of [[defining ingredient::braid group:B3]], i.e., the quotient of <math>B_3</math> by its [[center]].
 
* The [[defining ingredient::inner automorphism group]] of [[defining ingredient::braid group:B3]], i.e., the quotient of <math>B_3</math> by its [[center]].
 
* The [[defining ingredient::amalgamated free product]] of [[defining ingredient::cyclic group:Z4]] and [[defining ingredient::cyclic group:Z6]] over amalgamated subgroup [[cyclic group:Z2]] (living as [[Z2 in Z4]] and [[Z2 in Z6]] respectively).
 
* The [[defining ingredient::amalgamated free product]] of [[defining ingredient::cyclic group:Z4]] and [[defining ingredient::cyclic group:Z6]] over amalgamated subgroup [[cyclic group:Z2]] (living as [[Z2 in Z4]] and [[Z2 in Z6]] respectively).
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===Definition by presentation===
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The group can be defined by any of the following presentations (here, <math>e</math> denotes the identity element):
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* {{fillin}}
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* From the amalgamated free product definition: <math>\langle x,y \mid x^4 = e, x^2 = y^3 \rangle</math>
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===Structures===
 
===Structures===

Revision as of 16:45, 7 September 2012

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Definition

The group SL(2,\mathbb{Z}) is defined as the group, under matrix multiplication, of 2 \times 2 matrices over \mathbb{Z}, the ring of integers, having determinant 1.

In other words, it is the group with underlying set:

\left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad - bc = 1 \right \}

This is the degree two case of a special linear group over integers and hence of a special linear group. It is also a special case of a special linear group of degree two.

The group also has the following equivalent descriptions:

Definition by presentation

The group can be defined by any of the following presentations (here, e denotes the identity element):

  • PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
  • From the amalgamated free product definition: \langle x,y \mid x^4 = e, x^2 = y^3 \rangle


Structures

Thinking of SL(2,\mathbb{Z}) as a group of matrices, we see that it is an example of an arithmetic group.

Arithmetic functions

Function Value Explanation
order infinite (countable) As SL(2,\mathbb{Z}): The group is infinite because, for instance, it contains all matrices of the form \begin{pmatrix} 1 &  1 \\ 0 & 1 \\\end{pmatrix} for a \in \mathbb{Z}.
As a set, the group is contained in the set of all 2 \times 2 matrices over \mathbb{Z}. This can be identified with \mathbb{Z}^4, which is countable since \mathbb{Z} is countable. Thus, SL(2,\mathbb{Z}) is also countable.
As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite.
exponent infinite As SL(2,\mathbb{Z}): The group contains the element \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}, which has infinite order.
As the inner automorphism group of the braid group: infinite, because the braid group generators have infinite order.
minimum size of generating set 2 As SL(2,\mathbb{Z}): Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so SL(2,\mathbb{Z}) is generated by all matrices of the form \begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} 1 & 0 \\ b & 1 \\\end{pmatrix} with a,b varying over \mathbb{Z}. By the fact that the additive group of \mathbb{Z} is cyclic, all these matrices are generated by \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} and \begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}.
As the inner automorphism group of B_3: 2, because B_3 itself is 2-generated.
As an amalgamated free product of two cyclic groups: follows that it is 2, just from the definition (note that the generators here are different from those used in the justification in matrix terms).
subgroup rank infinite (countable) SL(2,\mathbb{Z}) has a subgroup that is isomorphic to free group:F2 (see Sanov subgroup in SL(2,Z) is free of rank two). This in turn has free subgroups of countable rank.

Group properties

Property Satisfied? Explanation Corollary properties satisfied/dissatisfied
2-generated group Yes See explanation for minimum size of generating set above satisfies: finitely generated group, countable group
Noetherian group No See explanation for subgroup rank above
finitely presented group Yes Any of the three definitions (SL, braid group inner automorphisms, amalgamated free product) gives a finite presentation
solvable group No contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two dissatisfies: nilpotent group, abelian group
group satisfying no nontrivial identity Yes contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two
SQ-universal group Yes contains subgroup isomorphic to free group:F2 -- see Sanov subgroup in SL(2,Z) is free of rank two
residually finite group Yes The kernels of the homomorphisms SL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/n\mathbb{Z}) for natural numbers n are normal subgroups of finite index and their intersection is trivial. satisfies: finitely generated residually finite group
Hopfian group Yes Follows from finitely generated and residually finite implies Hopfian satisfies: finitely generated Hopfian group

Facts

GAP implementation

Description Functions used
SL(2,Integers) SL, Integers