SL2Z

SL(2,Z) is the group of 2 by 2 matrices with integer entries and determinant 1. It is a discrete subgroup of SL(2,R) and is commonly called the modular group. The group acts by Möbius transformations on the extended complex plane and, in particular, on the upper half-plane, where it furnishes a rich structure used in number theory and geometry.

The standard generators are S = [0 -1; 1 0] and T = [1 1; 0 1]. In SL(2,Z) one has S^2 = -I and S^4 = I. The product ST equals [0 -1; 1 1], and (ST)^3 = -I, so (ST)^6 = I. Consequently, a convenient presentation for SL(2,Z) is given by generators S and T with the relations S^4 = I and (ST)^3 = -I (equivalently, (ST)^6 = I). The center of SL(2,Z) is {±I}.

The quotient PSL(2,Z) = SL(2,Z)/{±I} is isomorphic to the free product C2 * C3, reflecting that the images of S and ST have orders 2 and 3, respectively. This contrasts with SL(2,Z) itself, where ±I remains nontrivial in the center.

As a geometric action, SL(2,Z) preserves the tessellation of the upper half-plane by ideal triangles and has a fundamental domain commonly described as the region with |z| ≥ 1 and −1/2 ≤ Re(z) ≤ 1/2. The modular group and its relatives appear centrally in the theories of modular forms, elliptic curves, and various areas of arithmetic geometry and topology.