Modular group

In mathematics, the modular group is the projective special linear group ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$ of ${\displaystyle 2\times 2}$ matrices with integer coefficients and determinant ${\displaystyle 1}$, such that the matrices ${\displaystyle A}$ and ${\displaystyle -A}$ are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations. The name "modular group" comes from the relation to moduli spaces, and not from modular arithmetic.

The modular group Γ is the group of fractional linear transformations of the complex upper half-plane, which have the form

where ${\displaystyle a,b,c,d}$ are integers, and ${\displaystyle ad-bc=1}$. The group operation is function composition.

This group of transformations is isomorphic to the projective special linear group ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$, which is the quotient of the 2-dimensional special linear group ${\displaystyle \operatorname {SL} (2,\mathbb {Z} )}$ by its center ${\displaystyle \{I,-I\}}$. In other words, ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$ consists of all matrices

where ${\displaystyle a,b,c,d}$ are integers, ${\displaystyle ad-bc=1}$, and pairs of matrices ${\displaystyle A}$ and ${\displaystyle -A}$ are considered to be identical. The group operation is usual matrix multiplication.

Some authors define the modular group to be ${\displaystyle \operatorname {PSL} (2,\mathbb {Z} )}$, and still others define the modular group to be the larger group ${\displaystyle \operatorname {SL} (2,\mathbb {Z} )}$.

Some mathematical relations require the consideration of the group ${\displaystyle \operatorname {GL} (2,\mathbb {Z} )}$ of matrices with determinant plus or minus one. (${\displaystyle \operatorname {SL} (2,\mathbb {Z} )}$ is a subgroup of this group.) Similarly, ${\displaystyle \operatorname {PGL} (2,\mathbb {Z} )}$ is the quotient group ${\displaystyle \operatorname {GL} (2,\mathbb {Z} )/\{I,-I\}}$.

Since all ${\displaystyle 2\times 2}$ matrices with determinant 1 are symplectic matrices, then ${\displaystyle \operatorname {SL} (2,\mathbb {Z} )=\operatorname {Sp} (2,\mathbb {Z} )}$, the symplectic group of ${\displaystyle 2\times 2}$ matrices.