Special linear group:SL(2,Z)

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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:

$\{\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)\mid a,b,c,d\in \mathbb{Z},ad-bc=1\}$

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 inner automorphism group of braid group:B3, i.e., the quotient of $B_3$ by its center.
• 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).

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 $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$ 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 $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$, 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 $\left(\begin{array}{cc}1& a\\ 0& 1\\ \end{array}\right)$ and Failed to parse (syntax error): {\displaystyle 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 $\left(\begin{array}{cc}1& 1\\ 0& 1\\ \end{array}\right)$ and $\left(\begin{array}{cc}1& 0\\ 1& 1\\ \end{array}\right)$. 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.

GAP implementation

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