On the membership problem for some subgroups of $$SL_2(\mathbf {Z})$$ S
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On the membership problem for some subgroups of SL2 (Z) Henri-Alex Esbelin1
· Marin Gutan2
Received: 22 December 2017 / Accepted: 6 June 2018 © Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018
1λ and For λ ∈ Z, λ ≥ 2, consider the group G λ generated by two matrices Aλ = 01 10 . We present new algorithms for the membership problem for the groups G λ . Bλ = λ1 Let us consider another family of subgroups of S L 2 (Z) defined by Abstract
Gλ =
1 + λ2 n 1 λn 2 λn 3 1 + λ2 n 4
∈ S L 2 (Z) | (n 1 , n 2 , n 3 , n 4 ) ∈ Z4 .
Using continued fractions with partial quotients in λZ, we characterize the matrices of the group Gλ which belong to G λ . Our results are extensions of the classical result of I. Sanov for G 2 = G2 . Keywords Matrix groups · Free groups · Membership problem · Continued fractions Mathematics Subject Classification 20E05 · 20H05 · 11Y16 1λ et Bλ = Soient λ ∈ Z, λ ≥ 2, et G λ le groupe des matrices engendré par Aλ = 01 10 . Nous proposons de nouveaux algorithmes qui, sur la donée d’une matrice de S L 2 (Z), λ1 répond à la question de son appartenance à G λ . On considére une autre famille de sousgroupes de S L 2 (Z) définie par Résumé
Gλ =
B
1 + λ2 n 1 λn 2 λn 3 1 + λ2 n 4
∈ S L 2 (Z) | (n 1 , n 2 , n 3 , n 4 ) ∈ Z4 .
Henri-Alex Esbelin [email protected] Marin Gutan [email protected]
1
Present Address: Université Clermont Auvergne, Limos, 63177 Aubière Cedex, France
2
Université Clermont Auvergne, CNRS, LMBP, 63177 Aubière Cedex, France
123
H.-A. Esbelin, M. Gutan
En utilisant les fractions continues de quotients partiels dans λZ, nous caractérisons les matrices du groupe Gλ qui appartiennent à G λ . Ces résultats généralisent le résultat classique de I. Sanov qui dit que G 2 = G2 .
1 Introduction
1λ 10 and B = Bλ = . Let 01 λ1 G λ = Aλ , Bλ be the subgroup of S L 2 (Z) generated by Aλ and Bλ . In this manner we obtain a family of free subgroups of S L 2 (Z) (see Sanov [8] for the case λ = 2 and Lyndon and Ullmann [7] in the general case). Some authors call the groups G λ parabolic Möbius groups. Clearly G λλ ⊂ G λ ∩ G λ for every (λ, λ ) ∈ (N\{0, 1})2 . Let us consider another family of groups (Gλ )λ∈Z,λ≥2 . The groups Gλ are subgroups of S L 2 (Z) defined by For λ ∈ Z, λ ≥ 2, consider the matrices A = Aλ =
Gλ =
1 + λ2 n 1 λn 2 λn 3 1 + λ2 n 4
∈ S L 2 (Z) | (n 1 , n 2 , n 3 , n 4 ) ∈ Z4 .
Obviously G λ ⊆ Gλ . In [8] , Sanov proved that G 2 = G2 . For the case λ ≥ 3 no similar simple characterization of the elements of the group G λ was known until now. We give such a characterization in the last section of our paper (see Theorems 3 and 4). A. Chorna, K. Geller and V. Shpilrain proved in [3] that for λ ≥ 3 G λ ⊂ Gλ and the subgroup G λ has infinite index in Gλ . The subgroup membership problem for S L 2 (Z) is solvable (see [5]). The solution given in [5] is based on the automatic structure of S L 2 (Z). In [3] a nice algorithm which decides if a matrix M of S L 2 (Z) belongs to G λ
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