Continued Fractions for a Class of Triangle Groups

Calta, Kariane; Schmidt, Thomas A.

doi:10.1017/s1446788712000651

Cited by 9 publications

(8 citation statements)

References 29 publications

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“…Our algorithms extend previous work; the algorithm for the class containing the Hecke group ∆(2, 8, ∞) coincides with the octagon Farey map in [27], [26], which in turn is a "folded version" of the map in [1]. In [8] continued fractions -different from ours-generating the groups ∆(3, m, ∞) (which are related to the Veech surfaces in [31], [7, §6.1]) are introduced; these triangle groups are 2-arithmetic precisely for m = 4, 5, 6.…”

Section: Introductionsupporting

confidence: 60%

Decreasing height along continued fractions

Panti¹

2018

Ergod. Th. Dynam. Sys.

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The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x → x −1 − x −1 eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of noncocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that noncocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps.Our proof is based on an analysis of the action of nonnegative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

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Section: Introductionsupporting

confidence: 60%

Decreasing height along continued fractions

Panti¹

2018

Ergod. Th. Dynam. Sys.

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“…Also, all Hecke groups H q are included in generalized Hecke groups H p;q . Also, generalized Hecke groups H p;q have been studied extensively for many aspects in the literature (for examples, please see, [3], [4], [5], [6], [7] and [8]).…”

Section: Introductionmentioning

confidence: 99%

Some Normal Subgroups of Extended Generalized Hecke Groups

Demır¹,

Koruoğlu²

2016

HJMS

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Generalized Hecke group H p;1 () is generated by X(z) = (z p) 1 and Y (z) = (z +) 1 where p = 2 cos p ; p 2 integer and 2. Extended generalized Hecke group H p;1 () is obtained by adding the re ‡ection R(z) = 1=z to the generators of generalized Hecke group H p;1 (): In this paper, we study the commutator subgroups of extended generalized Hecke groups H p;1 (): Also, we determine the power subgroups of generalized Hecke groups H p;1 () and extended generalized Hecke groups H p;1 ().

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“…Furthermore all Hecke groups H q are included in generalized Hecke groups H p,q . Generalized Hecke groups H p,q have been also studied by Calta and Schmidt in [2] and [3].…”

Section: S(z)mentioning

confidence: 99%

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

Kaymak

Demır

Koruoğlu

et al. 2018

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 and let Hp,q be the generalized Hecke group associated to p and q: The generalized Hecke group Hp,q is generated by X(z) = -(z-λp)-1 and Y (z) = -(z+ λq)-1 where λp = 2 cos ≤ π/p and λq = 2 cos π/q . The extended generalized Hecke group H̅p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group Hp,q: In this paper, we study the commutator subgroups of generalized Hecke groups Hp,q and extended generalized Hecke groups H̅p,q.

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Continued Fractions for a Class of Triangle Groups

Cited by 9 publications

References 29 publications

Decreasing height along continued fractions

Decreasing height along continued fractions

Some Normal Subgroups of Extended Generalized Hecke Groups

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

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