Solving the membership problem for parabolic Möbius monoids

• Published in: Semigroup forum 2019-06, Vol.98 (3), p.556-570, Article 556
• Main Authors: Esbelin, Henri-Alex, Gutan, Marin
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Computer Science; Discrete Mathematics; Mathematics; Mathematics and Statistics; Monoids; Quotients; Research Article; Subgroups
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-019-10013-4

For λ ∈ Q , λ > 0 , consider the submonoid S λ (resp. the subgroup G λ ) of S L 2 ( Q ) generated by two parabolic matrices A λ = 1 λ 0 1 and B λ = 1 0 λ 1 . We present new results and algorithms for the membership problem for the monoids S λ . For λ ∈ N ∗ , let us also consider the families of monoids and subgroups of S L 2 ( Z ) defined by S λ = 1 + λ 2 n 1 λ n 2 λ n 3 1 + λ 2 n 4 ∈ S L 2 ( Z ) ∣ ( n 1 , n 2 , n 3 , n 4 ) ∈ N 4 , G λ = 1 + λ 2 n 1 λ n 2 λ n 3 1 + λ 2 n 4 ∈ S L 2 ( Z ) ∣ ( n 1 , n 2 , n 3 , n 4 ) ∈ Z 4 . Using continued fractions with partial quotients in λ N , we characterize the matrices of the monoid S λ which belong to S λ . Our results are analogues for monoids of the classical result for groups of I. Sanov which says that G 2 = G 2 .