Special linear group:SL(2,Z)

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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, 1 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 = 1, 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 &  a \\ 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.
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 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 definitions (SL, 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 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

Elements

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


Facts

GAP implementation

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