Farey Symbols and Fundamental Domains
One method of finding fundamental domains for congruence subgroups is the method of Farey Symbols, as described by Kulkarni [Kul91].
A generalized Farey sequence is a sequence of rationals
such that for a consecutive pair of fractions \({b \over d}, {a \over c}\) in the sequence, written in lowest terms, we have \(ad - bc = 1\). We extend the rationals to \({\mathbb{Q}}\cup\{-\infty,\infty\}\), where we use the convention \(-\infty = {-1\over 0}\) and \(\infty = {1\over 0}\).
A Farey Symbol is a Farey sequence of length \(n\) starting with \({-1\over 0}\), and ending with \({1\over 0}\), together with a sequence of \(n-1\) labels. We use the convention that labels can be any elements of \({\mathbb{N}}_{>0}\cup\{-2,-3\}\). The sequence of labels must satisfy the condition that each element of \({\mathbb{N}}_{>0}\) appears in the sequence either exactly twice or not at all. For example, the sequences \(\left[{-1\over 0},{0\over 1}, {1\over 2},{2\over 3},{5\over 7},{3\over 4},{1\over 1},{1\over 0}\right]\) and \(\left[ 1,2,2,-3,-2,-2,1\right]\) define a Farey symbol, which is generally written in the following format:
Farey symbols are used to define certain fundamental domains for congruence subgroups of \({\operatorname{PSL}}_2({\mathbb{Z}})\). The sequence of fractions gives cusps which are vertices of the domain, and the labels give edge identifications. For \(a_i,a_{i+1}\) in the Farey sequence, with corresponding label \(l_i\) not \(-3\), the corresponding edge of the domain is a geodesic between \(a_i\) and \(a_{i+1}\). If the label is \(-3\), there is an extra elliptic point of order \(3\) on the boundary of the domain between the two cusps, and the two edges between these cusps are identified. The label \(l_i=-2\) indicates an elliptic point of order \(2\) on the boundary between the two cusps \(a_i\) and \(a_{i+1}\). This point is on the geodesic between \(a_i\) and \(a_{i+1}\), and the two halves of the geodesic are identified.
- FareySymbol(G): GrpPSL2 -> SymFry
Computes the Farey Symbol of a congruence subgroup \(G\) in \({\operatorname{PSL}}_2({\mathbb{Z}})\).
- Cusps(FS): SymFry -> SeqEnum
Returns the cusp sequence of the Farey symbol \(FS\). Note, this is not a sequence of inequivalent cusps of the corresponding group.
- Generators(FS): SymFry -> SeqEnum
Returns the generators of the congruence subgroup corresponding to the Farey symbol \(FS\).
- Group(FS): SymFry -> GrpPSL2
Returns the congruence subgroup corresponding to the Farey Symbol \(FS\).
- Widths(FS): SymFry -> SeqEnum
Returns the sequence of integers giving twice the widths of the cusp list of the Farey symbol \(FS\).
- Index(FS): SymFry -> RngIntElt
Returns the index of
Group(FS)in \({\operatorname{PSL}}_2({\mathbb{Z}})\).
- FundamentalDomain(FS): SymFry -> SeqEnum
- FundamentalDomain(FS, H): SymFry, SpcHyp -> SeqEnum
Returns the vertices in the upper half plane of the fundamental domain described by the Farey Sequence \(FS\). A second argument may be given to specify the upper half plane \(H\).