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

Latest revision as of 20:47, 18 September 2012

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The group $S L (2, Z)$ is defined as the group, under matrix multiplication, of $2 \times 2$ matrices over $Z$ , the ring of integers, having determinant $1$ .

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

${(\begin{matrix} a & b \\ c & d \end{matrix}) ∣ a, b, c, d \in Z, a d - b c = 1}$

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:

The amalgamated free product of cyclic group:Z4 and cyclic group:Z6 over amalgamated subgroup cyclic group:Z2 (living as Z2 in Z4 and Z2 in Z6 respectively).

Definition by presentation

The group can be defined by any of the following presentations (here, $1$ denotes the identity element):

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From the amalgamated free product definition: $⟨ x, y ∣ x^{4} = 1, x^{2} = y^{3} ⟩$

Structures

Thinking of $S L (2, Z)$ 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 $S L (2, Z)$ : The group is infinite because, for instance, it contains all matrices of the form $(\begin{matrix} 1 & a \\ 0 & 1 \end{matrix})$ for $a \in Z$ . As a set, the group is contained in the set of all $2 \times 2$ matrices over $Z$ . This can be identified with $Z^{4}$ , which is countable since $Z$ is countable. Thus, $S L (2, Z)$ 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 $S L (2, Z)$ : The group contains the element $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ , which has infinite order.
minimum size of generating set	2	As $S L (2, Z)$ : Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so $S L (2, Z)$ is generated by all matrices of the form $(\begin{matrix} 1 & a \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} 1 & 0 \\ b & 1 \end{matrix})$ with $a, b$ varying over $Z$ . By the fact that the additive group of $Z$ is cyclic, all these matrices are generated by $(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix})$ . 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)	$S L (2, Z)$ 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 definitions ( $S L$ , B_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 $S L (2, Z) \to S L (2, Z / n Z)$ 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

Elements

Further information: element structure of special linear group:SL(2,Z)

Facts

Sanov subgroup in SL(2,Z) is free of rank two
The homomorphism $S L (2, Z) \to S L (2, Z / n Z)$ is surjective for any natural number $n$ .

GAP implementation

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

@@ Line 13: / Line 13: @@
 The group also has the following equivalent descriptions:
-* 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).
 ===Definition by presentation===
-The group can be defined by any of the following presentations (here, <math>e</math> denotes the identity element):
+The group can be defined by any of the following presentations (here, <math>1</math> denotes the identity element):
 * {{fillin}}
-* From the amalgamated free product definition: <math>\langle x,y \mid x^4 = e, x^2 = y^3 \rangle</math>
+* From the amalgamated free product definition: <math>\langle x,y \mid x^4 = 1, x^2 = y^3 \rangle</math>
 ===Structures===
@@ Line 33: / Line 31: @@
 ! Function !! Value !! Explanation
 |-
-| [[order of a group|order]] || infinite (countable) || As <math>SL(2,\mathbb{Z})</math>: The group is infinite because, for instance, it contains all matrices of the form <math>\begin{pmatrix} 1 &  1 \\ 0 & 1 \\\end{pmatrix}</math> for <math>a \in \mathbb{Z}</math>. <br>As a set, the group is contained in the set of all <math>2 \times 2</math> matrices over <math>\mathbb{Z}</math>. This can be identified with <math>\mathbb{Z}^4</math>, which is countable since <math>\mathbb{Z}</math> is countable. Thus, <math>SL(2,\mathbb{Z})</math> is also countable.<br>As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite.
+| [[order of a group|order]] || infinite (countable) || As <math>SL(2,\mathbb{Z})</math>: The group is infinite because, for instance, it contains all matrices of the form <math>\begin{pmatrix} 1 &  a \\ 0 & 1 \\\end{pmatrix}</math> for <math>a \in \mathbb{Z}</math>. <br>As a set, the group is contained in the set of all <math>2 \times 2</math> matrices over <math>\mathbb{Z}</math>. This can be identified with <math>\mathbb{Z}^4</math>, which is countable since <math>\mathbb{Z}</math> is countable. Thus, <math>SL(2,\mathbb{Z})</math> is also countable.<br>As an amalgamated free product: any amalgamated free product relative to a subgroup that is proper in both groups is infinite.
 |-
-| [[exponent of a group|exponent]] || infinite  || As <math>SL(2,\mathbb{Z})</math>: The group contains the element <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, which has infinite [[order of an element|order]].<br>As the inner automorphism group of the braid group: infinite, because the braid group generators have infinite order.
+| [[exponent of a group|exponent]] || infinite  || As <math>SL(2,\mathbb{Z})</math>: The group contains the element <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math>, which has infinite [[order of an element|order]].
 |-
-| {{arithmetic function value|minimum size of generating set|2}} || As <math>SL(2,\mathbb{Z})</math>: Follows from [[elementary matrices of the first kind generate the special linear group over a Euclidean ring]], so <math>SL(2,\mathbb{Z})</math> is generated by all matrices of the form <math>\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ b & 1 \\\end{pmatrix}</math> with <math>a,b</math> varying over <math>\mathbb{Z}</math>. By the fact that the additive group of <math>\mathbb{Z}</math> is cyclic, all these matrices are generated by <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}</math>.<br>As the inner automorphism group of <math>B_3</math>: 2, because <math>B_3</math> itself is 2-generated. <br>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).
+| {{arithmetic function value|minimum size of generating set|2}} || As <math>SL(2,\mathbb{Z})</math>: Follows from [[elementary matrices of the first kind generate the special linear group over a Euclidean ring]], so <math>SL(2,\mathbb{Z})</math> is generated by all matrices of the form <math>\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ b & 1 \\\end{pmatrix}</math> with <math>a,b</math> varying over <math>\mathbb{Z}</math>. By the fact that the additive group of <math>\mathbb{Z}</math> is cyclic, all these matrices are generated by <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> and <math>\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}</math>.<br>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) || <math>SL(2,\mathbb{Z})</math> 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.
@@ Line 51: / Line 49: @@
 | [[dissatisfies property::Noetherian group]] || No || See explanation for subgroup rank above ||
 |-
-| [[satisfies property::finitely presented group]] || Yes || Any of the three definitions (<math>SL</math>, braid group inner automorphisms, amalgamated free product) gives a finite presentation ||
+| [[satisfies property::finitely presented group]] || Yes || Any of the definitions (<math>SL</math>, B_amalgamated free product) gives a finite presentation ||
 |-
 | [[dissatisfies property::solvable group]] || No || contains subgroup isomorphic to [[free group:F2]] -- see [[Sanov subgroup in SL(2,Z) is free of rank two]] || dissatisfies: [[dissatisfies property::nilpotent group]], [[dissatisfies property::abelian group]]
@@ Line 63: / Line 61: @@
 | [[satisfies property::Hopfian group]] || Yes || Follows from [[finitely generated and residually finite implies Hopfian]] || satisfies: [[satisfies property::finitely generated Hopfian group]]
 |}
+==Elements==
+{{further|[[element structure of special linear group:SL(2,Z)]]}}
+{{#lst:element structure of special linear group:SL(2,Z)|summary}}
 ==Facts==