Subgroups of PSL(2, R)

The group $GL^{+}_2(\mbox{\bf R})$ of 2 by 2 matrices defined over $\mbox{\bf R}$ with positive determinant acts on the upper half complex plane $\mbox{\tt H}=\{x\in C \vert \mbox{\rm Im}(x) > 0\}\}$ by fractional linear transformation:

$$\pmatrix{a & b\cr c & d}:z \mapsto {az + b \over cz+d}.$$

Any subgroup $\Gamma$ of $GL^{+}_2(\mbox{\bf R})$ also acts on $\mbox{\tt H}$. A fundamental domain for the action of $\Gamma$ is a region of $\mbox{\tt H}^*$ containing a representative of each orbit of the action. Magma contains a package written by Helena Verrill for working with $\mbox{\tt H}^*$ and with congruence subgroups and their action on $\mbox{\tt H}^*$. The subgroups of $\mbox{\rm PSL}_2(Z)$ currently allowed are those of the form $\Gamma_0(N)$, $\Gamma_1(N)$, $\Gamma(N)$, $\Gamma^1(N)$, $\Gamma^0(N)$, and intersections of these groups. The package allows the computation of generators for congruence subgroups, and various other information, such as coset representatives.

• Computation of generators of congruence subgroups
• Coset representatives for a subgroup of finite index in $\mbox{\rm PSL}_2(\mbox{\bf Z})$
• Construction of cusps, cusp widths, and elliptic points of congruence subgroups
• Farey symbols for congruence subgroups
• Action of elements of $\mbox{\rm PSL}_2(\mbox{\bf R})$ on the upper half complex plane
• Determination of vertices of a fundamental domain for the action of a congruence subgroup
• Equivalence of points under the action of a congruence subgroup
• Graphics: postscript output of pictures of fundamental domains, points and geodesics, and polygons with geodesic edges (all on the upper half complex plane)