ModularClasses

Every hyperbolic conjugacy class whose word has the given length — equivalently, every closed geodesic of that symbolic period on the modular surface. Conjugation rotates the word, so the classes are binary NECKLACES.

WikipediaModular group↗MathWorldModularGroupGamma↗WikidataQ1834342↗

ModularClasses(length)the classes, each named by the least rotation of its word

ModularClasses(length, True)only the primitive ones — not a repeat of a shorter geodesic

ModularClass(word)the canonical name of one class

Domain: The modular group

Details

• Conjugating by the first letter rotates the word:
• The count is — binary necklaces, less the two constant ones, which are parabolic
• The primitive classes are the aperiodic necklaces, counted by — Lyndon words
• Word length is the SYMBOLIC period; the geodesic's length is , and the two orderings differ

Examples

See also: RademacherSymbol, ModularWord