Farey boat I. Continued fractions and triangulations, modular group and polygon dissections
aa r X i v : . [ m a t h . C O ] J a n FAREY BOAT: CONTINUED FRACTIONS AND TRIANGULATIONS,MODULAR GROUP AND POLYGON DISSECTIONS
SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
Abstract.
We reformulate several known results about continued fractions in combinatorial terms.Among them the theorem of Conway and Coxeter and that of Series, both relating continued frac-tions and triangulations. More general polygon dissections appear when extending these theorems forelements of the modular group PSL(2 , Z ). These polygon dissections are interpreted as walks in theFarey tessellation. The combinatorial model of continued fractions can be further developed to obtaina canonical presentation of elements of PSL(2 , Z ). `A la m´emoire de Christian Duval Introduction
In this paper we formulate combinatorial interpretations of algebraic properties of continued fractionsand of matrices in the modular group PSL(2 , Z ). The combinatorics is related to polygon dissections andwalks in the Farey tessellation.The starting point of the present paper is a theorem due to Conway and Coxeter [11]. This theoremuses triangulations of polygons to classify Coxeter’s “frieze patterns”. Work of Coxeter on frieze patternswas motivated by continued fractions; see [12]. Our first goal is to reformulate Conway and Coxeter’stheorem (and related notions) directly in terms of continued fractions, and to compare it to some knownresults in the area. In particular, we compare the Conway and Coxeter theorem with the theorem ofSeries [33] that provides an embedding of continued fractions into the Farey tessellation. This comparisonoffers a combinatorial relation between negative and regular continued fractions.The second goal of the paper is to develop the combinatorics that arose from the above comparison.This leads to surprising results and notions, that appeared recently in the literature [9, 31]. Amongthem are relationship between continued fractions and Pfaffians of skew-symmetric matrices, and tosome particular polygon dissections. We give a survey of this recent development. Furthermore, alongthe same lines, we obtain several statements that appear to be new. These are Theorems 4.13, 5.5, 5.7and 6.3. We understand elements of PSL(2 , Z ) as generalized (finite) continued fractions, triangulationsare replaced by more general polygon dissections.Let us outline possible applications and further developments of the combinatorial approach discussedin this paper. We believe that the relation between PSL(2 , Z ) and polygon dissections (see, in particular,Theorems 3.7 and 6.3) can be applied to other groups extending PSL(2 , Z ). This relation connects thetopic with several other areas of algebra, geometry and combinatorics (such as cluster algebras, friezepatterns, etc.). Application and combination of various methods known in these areas look promising.One application is already explored in the second part of this work [28], where we suggest a notion of q -deformed continued fractions and of q -deformed rational numbers; this deformation preserves the com-binatorial properties discussed in the present paper.The paper consists of six sections, each of them can be read independently. Key words and phrases.
Continued fractions, Farey graph, polygon dissections, Ptolemy rule, Pfaffians, modular group.
SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
In Section 1 we expose a combinatorial model for continued fractions. We consider two classicallyknown expansions of a rational number rs = c − c − . . . − c k = a + 1 a + 1 . . . + 1 a m , with c i ≥ a i ≥
1. The algebraic relationship between these two expansions due to Hirzebruch [21]is encoded in a triangulation of a polygon. Although this section is introductory, it contains the maintools used throughout the paper, such as Ptolemy rule and triangulations of polygons in the Farey graph.The statements in this section are essentially reformulations of results that can be found in terms offrieze patterns in Coxeter [12] and results in terms of hyperbolic geometry in Series [33]. We call thesestatements “Facts” and illustrate them on running examples.In Section 2 we focus on the matrices(0.1) M ( c , . . . , c k ) := c −
11 0 ! c −
11 0 ! · · · c k −
11 0 ! and M + ( a , . . . , a m ) := a
11 0 ! a
11 0 ! · · · a m
11 0 ! associated with the continued fractions. We establish elementary algebraic properties of these matricesand in particular their algebraic relationship. In this section, the remarkable identity M ( c , . . . , c n ) = − Idappears using the combinatorial data introduced in Section 1.In Section 3 we describe combinatorially the complete set of positive integer n -tuples ( c , . . . , c n ) thatare solutions of the equation(0.2) M ( c , . . . , c n ) = ± Id . The theorem of Conway and Coxeter [11] provides a certain subset of solutions of M ( c , . . . , c n ) = − Idin terms of triangulations of n -gons. These solutions are obtained from the triangulations by countingthe number of triangles incident at each vertex of the n -gon. All positive integer solutions of (0.2) areobtained from a special class of dissections of n -gons called “3 d -dissections” [31]. Under weaker conditionson the coefficients c i , the continued fraction disappears gradually, but the corresponding combinatoricslives on and becomes more sophisticated.Let us be a little bit more technical and briefly explain the way combinatorics appears in the contextof the modular group. This relationship is central for the whole paper. The standard choice of generatorsof PSL(2 , Z ) is R = ! , S = −
11 0 ! , and all the relations in PSL(2 , Z ) are consequences of the following two relations: S = Id and ( RS ) = Id,implying the well-known isomorphism PSL(2 , Z ) ≃ ( Z / Z ) ∗ ( Z / Z ). It is then not difficult to deducethat every A ∈ PSL(2 , Z ) can be written (non-uniquely) in the form(0.3) A = R c S R c S · · · R c n S, in such a way that all the coefficients c i are positive integers . Note that (0.3) coincides with (0.1),i.e., A = M ( c , . . . , c n ). It is a rule in combinatorics that positive integers count some objects, andTheorem 3.7 provides this interpretation in the case where A is a relation in PSL(2 , Z ), i.e., when A = Id. ...similar to Cheshire cat’s grin. AREY BOAT 3
In Section 4 solutions of (0.2) are embedded into the Farey graph. The embedding makes use of thesequence of rationals defined as the convergents of the negative continued fraction corresponding to apositive solution. Conway and Coxeter’s solutions are then identified with “Farey polygons” (as provedin [29]). More general solutions correspond to “walks on Farey polygons”.In Section 5 we connect the topic to the Ptolemy-Pl¨ucker relations (and thus to cluster algebras;see [14, 15]). The origin of these considerations goes back to Euler who proved a series of identitiesfor the “continuants”, i.e., the polynomials describing continued fractions in terms of the coefficients c i (or a i ). Following [34], we interpret Euler’s identity in terms of the Pfaffian of a 4 × M ( c , . . . , c n ).Note that this appearance of Pfaffians is not a simple artifact, it reflects a relationship between thesubject and symplectic geometry; see [10]. However, we do not describe this relationship in the presentpaper.Section 6 formulates some consequences of the developed combinatorics for the modular group PSL(2 , Z ).Every element A of PSL(2 , Z ) can be written in the form A = M ( c , . . . , c k ) in infinitely many differentways. We make such a presentation canonical by imposing the conditions c i ≥ k being the smallestpossible, and deduce presentations of A in the standard generators of PSL(2 , Z ). We prove that thecanonical presentation A = M ( c , . . . , c k ) is given by the expansion into the negative continued fractionof the quotient of greatest coefficients of A . Matrices M ( c , . . . , c k ) with c i ≥ , Z ); see [22, 35, 24], the sequence ( c , . . . , c k ) being the period of the continuedfraction of a fixed point of A , which is a quadratic irrational. In our approach the quadratic irrational isreplaced by a rational number. Contents
Introduction 11. Continued fractions and triangulations 41.1. Triangulations with two exterior triangles 41.2. Combinatorial interpretation of continued fractions 51.3. The mirror formula 61.4. Farey sums and the labeling of vertices 71.5. Recovering r and s with the Ptolemy-Pl¨ucker rule 71.6. Triangulations T r/s inside the Farey graph 82. Matrices of negative and regular continued fractions 102.1. The matrices of continued fractions 102.2. Matrices M + ( a , . . . , a m ) and M ( c , . . . , c k ) in terms of the generators 112.3. Converting the matrices 112.4. Converting the conjugacy classes in PSL(2 , Z ) 122.5. Appearance of the equation M ( c , . . . , c n ) = − Id 122.6. The semigroup Γ 133. Solving the equation M ( c , . . . , c n ) = ± Id 133.1. Conway and Coxeter totally positive solutions 143.2. The complete set of solutions: 3 d -dissections 153.3. The total sum c + · · · + c n M ( c , . . . , c n ) = − Id and n -cycles in the Farey graph 174.2. Farey n -gons and triangulations 194.3. 3 d -dissections and walks on the Farey tessellation 204.4. The quiddity of a 3 d -dissection from a Farey walk 235. PPP: Ptolemy, Pl¨ucker and Pfaff 23 SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO d -dissections 275.5. Traces and Pfaffians 276. Minimal presentation of PSL(2 , Z ) 286.1. Parametrizing the conjugacy classes in PSL(2 , Z ) 296.2. Minimal presentation 296.3. Proof of Theorem 6.3 and Proposition 6.4 316.4. Minimal presentation and Farey n -gon 326.5. Further examples: Cohn matrices 336.6. The 3 d -dissection of a matrix 34References 351. Continued fractions and triangulations
This section is a collection of basic properties of continued fractions that we formulate in a combina-torial manner.Let r and s be two coprime positive integers, and assume that r > s . The rational number rs hasunique expansions(1.1) rs = c − c − . . . − c k = a + 1 a + 1 . . . + 1 a m , where c i ≥ a i ≥
1, for all i .The first expansion is usually called a negative , or reversal continued fraction the second is a (morecommon) regular continued fraction . We will use the notation J c , . . . , c k K and [ a , . . . , a m ] for the abovecontinued fractions, respectively. Note that one can always assume the number of terms in the regularcontinued fraction to be even, since [ a , . . . , a ℓ + 1] = [ a , . . . , a ℓ , c , . . . , c k ) in terms of the coefficients ( a , . . . , a m ),whenever J c , . . . , c k K = [ a , . . . , a m ], is as follows:(1.2) ( c , . . . , c k ) = (cid:0) a + 1 , , . . . , | {z } a − , a + 2 , , . . . , | {z } a − , . . . , a m − + 2 , , . . . , | {z } a m − (cid:1) . This expression can be found in [21, Eq. (19), p.241] and [22, Eqs. (22), (23)], see also [6, p.93]. We willgive a combinatorial explanation of this formula. In Section 2.3 we will give a detailed proof of a moregeneral statement.The goal of this introductory section is to explain that both, regular and negative, continued fractionscan be encoded by the same simple combinatorial picture. We will be considering triangulated n -gonswith exactly two exterior triangles. All the statements of this section are combinatorial reformulationsof known results.1.1. Triangulations with two exterior triangles.
Given a (convex) n -gon, we will be considering theclassical notion of triangulation which is a maximal dissection of the n -gon by diagonals that never crossexcept for the endpoints. A triangle in a triangulation is called exterior if two of its sides are also sides(and not diagonals) of the n -gon. AREY BOAT 5
In this section, we consider only those triangulations that have exactly two exterior triangles. In sucha triangulation the diagonal connecting the exterior vertices of the exterior triangles has the property tocross every diagonal of the triangulation: • ♣♣♣♣♣♣♣♣ ❂❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ • ❂❂❂❂❂❂❂❂❂❂ • • ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ ❴❴❴❴❴❴❴❴ • ◆◆◆◆◆◆◆◆ • ◆◆◆◆◆◆◆◆ /o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o/o • ♣♣♣♣♣♣♣♣ • • • • • ❴❴❴❴ • Then, every triangle in the triangulation (except for the exterior ones) can be situated with respect tothis diagonal in one of the two possible ways: • ✎✎✎✎✎✎ ✴✴✴✴✴✴ • • /o/o/o/o/o/o/o/o/o/o /o/o/o/o/o/o/o/o/o/o • • • ✴✴✴✴✴✴ ✎✎✎✎✎✎ that we refer to as “base-down” or “base-up”. We assume the first exterior triangle to be situatedbase-down, and the last one base-up.We enumerate the vertices from 0 to n − ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ❂❂❂❂❂❂❂❂❂❂❂ ✁✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂❂ ❴❴❴❴❴❴❴❴ k k + 1 ✁✁✁✁✁✁✁✁✁ n − n − ❴❴❴❴❴❴❴ k + 2so that the exterior vertices are 0 and k + 1.1.2. Combinatorial interpretation of continued fractions.
Given an n -gon and its triangulationwith two exterior triangles, we fix the following notation.(1) The integers ( a , a , . . . , a m ) count the number of equally positioned triangles, i.e. the triangu-lation consists of the concatenation of a triangles base down, followed by a triangles base upand so on: c ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ s s a + + c ❂❂❂❂❂❂❂❂❂❂ c • ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂❂ u u a ( ( • ❴❴❴❴❴❴❴❴ c k c k +1 ✁✁✁✁✁✁✁✁✁✁ c k k a c n − c n − • i i a • ❴❴❴❴❴❴❴ c k +2 (2) The integers ( c , c , . . . , c n = c ) count the number of triangles at each vertex, i.e., the integer c i is the number of triangles incident to the vertex i .Formula (1.2) is equivalent to the fact that these sequences define the same rational number. Fact 1. If ( a , . . . , a m ) and ( c , . . . , c k ) are the integers defined by (1) and (2), respectively, then theyare the coefficients of the expansions of the same rational number as a regular and negative continuedfraction, i.e., [ a , . . . , a m ] = J c , . . . , c k K . SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
For a proof, see Section 2.3.It is clear that each of the data ( a , . . . , a m ) and ( c , . . . , c k ) defines uniquely (the same) triangulationof a polygon with two exterior triangles. The number n of vertices is related to the sequences via a + a + · · · + a m = n − , c + c + · · · + c k = n + k − . Fact 1 then implies the following.
Corollary 1.1.
The set of rationals rs > is in a one-to-one correspondence with triangulations ofpolygons with two exterior triangles. Definition 1.2.
Given a rational number rs >
1, we denote by T r/s the corresponding triangulation withtwo exterior triangles. Example 1.3.
One has 75 = [1 , , ,
1] = J , , K . The corresponding triangulation T / is • ✁✁✁✁✁✁✁✁✁✁ • ✁✁✁✁✁✁✁✁✁✁ • ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣✁✁✁✁✁✁✁✁✁✁ • ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ • • • The mirror formula.
Consider the reversal of a regular continued fraction: [ a m , a m − , . . . , a ],which is important in number theorey; see, e.g., [1].For every ℓ ≥
0, define the ℓ th convergent of the regular continued fraction [ a , . . . , a m ] by r ℓ s ℓ := a + 1 a + 1 . . . + 1 a ℓ . The convergents of the negative continued fraction are defined in a similar way.The following statement is known as the “mirror formula”: r m r m − = [ a m , a m − , . . . , a ] . In Section 2.1, we will prove this statement with the help of the matrix form of continued fractions.The conversion into a negative continued fraction resorts to the coefficients c i on the opposite verticesof T r/s . Corollary 1.4.
One has [ a m , a m − , . . . , a ] = J c k +2 , c k +3 , . . . , c n − K . Proof.
This formula follows from Fact 1 when “rotating” the triangulation T r/s . (cid:3) Example 1.5.
The reversal of the continued fraction from Example 1.3 is as follows:74 = [1 , , ,
1] = J , K . AREY BOAT 7
Farey sums and the labeling of vertices.
The rational rs can be recovered from the triangula-tion T r/s by an additive rule.Let us label the vertices of the n -gon by elements of the set Q ∪ (cid:8) (cid:9) . We start from and atvertices 0 and 1, respectively. We then extend this labeling to the whole n -gon by the following “Fareysummation formula”. Whenever two vertices of the same triangle have been assigned the rationals r ′ s ′ and r ′′ s ′′ , then the third vertex receives the label r ′ s ′ ⊕ r ′′ s ′′ := r ′ + r ′′ s ′ + s ′′ . This process is illustrated by the following example.(1.3) ✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ❂❂❂❂❂❂❂❂❂
72 103 ✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂ ❴❴❴❴❴❴❴❴ rs ✁✁✁✁✁✁✁✁✁✁✁
01 11 21 31 134 237 ❴❴❴❴❴❴❴❴
The following statement is easily proved by induction. It can be viewed as a reformulation of the resultof Series [33]; for more details, see Section 1.6.
Fact 2.
Labeling the vertices of the triangulation T r/s according to the above rule, the vertex k +1 receivesthe label rs . Remark 1.6.
More generally, all the rationals labeling the vertices 2 , , . . . , k, k + 1 are the consecutiveconvergents of the negative continued fraction J c , . . . , c k K representing rs .1.5. Recovering r and s with the Ptolemy-Pl¨ucker rule. In Euclidean geometry, the Ptolemyrelation is the formula relating the lengths of the diagonals and sides of an inscribed quadrilateral. Itreads x , x , = x , x , + x , x , , where x i,j is the Euclidean length between the vertices i and j . bb b b Gr ,n , so that they are often called Ptolemy-Pl¨ucker relations.We will use this name in the sequel. They became an important and general rule in the theory of clusteralgebras [14, 15]. The “Ptolemy-Pl¨ucker rule” provides a way to calculate new variables from the oldones.Let us explain how the Ptolemy rule allows one to calculate the numerator r and the denominator s of the continued fraction (1.1) from the corresponding triangulation T r/s .Given a triangulated n -gon with exactly two exterior triangles, we will assign a value x i,j to all itsedges ( i, j ) with i ≤ j , so that the system of equations(1.4) ( x i,j x k,ℓ = x i,k x j,ℓ + x i,ℓ x k,j , i ≤ k ≤ j ≤ ℓ,x i,i = 0 , is satisfied. The system (1.4) will be called the Ptolemy-Pl¨ucker relations . SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
Fact 3. (i) The labels x i,j satisfying (1.4) are uniquely determined by the values x i,j of the sides anddiagonals of the triangulation.(ii) Assume that x i,j = 1 whenever ( i, j ) is a side or a diagonal of the triangulation • ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ • ❂❂❂❂❂❂❂❂❂❂ • ❴❴❴❴ • • • • ❴❴❴❴ Then all the labels x i,j are positive integers.(iii) In the triangulation T r/s , the assumption from Part (ii) implies the labeling ( x ,k +1 = rx ,k +1 = s. Parts (i) and (ii) are widely known in the theory of cluster algebra; see [16, Section 2.1.1]. We do notdwell on the proof here.Part (iii) was already known to Coxeter [12, Eq.(5.6)] who proved (in a different context) the followingmore general statement.
Fact 4.
Under the assumption that x i,j = 1 whenever ( i, j ) is a side or a diagonal of the triangulation,the integers x i,j are calculated as × determinants: (1.5) x i,j = det r i r j s i s j ! = r i s j − s i r j , where r i s i and r j s j are the rationals labeling the vertices i and j , as in (1.3). We will prove yet a more general result in Section 5.3 (see Theorem 5.5).We illustrate the statement (iii) by the following diagram. • ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ s
1q 1q 1q 1q 1q 1q 1q 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p 0p /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o .n .n .n .n .n .n .n .n .n .n .n .n .n .n .n .n -m -m -m -m -m -m -m • ❂❂❂❂❂❂❂❂❂❂ • • ✁✁✁✁✁✁✁✁✁✁ ❂❂❂❂❂❂❂❂❂❂ • ❴❴❴❴❴❴❴❴ • • ✁✁✁✁✁✁✁✁✁✁ • r • • • • • ❴❴❴❴❴❴❴❴ • Triangulations T r/s inside the Farey graph. The triangulation (1.3) can be naturally embeddedin the Farey tessellation. In this section we explain how to extract the triangulation T r/s from the Fareytessellation. This construction is due to C. Series [33], and it allows one to deduce Fact 2 from her result. Definition 1.7. a) The set of all rational numbers Q , completed by ∞ represented by , form a graphcalled the Farey graph . Two rationals written as irreducible fractions, r ′ s ′ and r ′′ s ′′ , are connected by anedge if and only if r ′ s ′′ − r ′′ s ′ = ± Q ∪ {∞} into the border of the hyperbolic half-plane H , the edges are often representedas geodesics of H (which is a half-circle) and the Farey graph splits H into an infinite set of trianglescalled the Farey tessellation .Basic properties of the Farey graph and Farey tessellation can be found in [19], we will need thefollowing.a) The edges of the Farey tessellation never cross, except at the endpoints.b) Every triangle in the Farey graph is of the form n r ′ s ′ , r ′ + r ′′ s ′ + s ′′ , r ′′ s ′′ o , AREY BOAT 9 bbbbb b b b b b b b b b b b b b
01 11 54 43 75 32 85 53 74 21 73 52 83 31 72 41 51 10
Figure 1.
A fragment of the Farey graph r ′ s ′ r ′ + r ′′ s ′ + s ′′ r ′′ s ′′ We focus on the part of the graph consisting of rational numbers greater than 1.The construction of [33] is as follows. Fix a rational number rs and draw a vertical line ( L ) ⊂ H through rs . Collect all the triangles of the Farey tessellation crossed in their interior by this line. Thisleads to the triangulation T r/s .The property of the triangles to be situated “base down” and “base up” now read as: ”base at theleft of ( L )” and ”base at the right of ( L )”. The two exterior vertices are and rs . The vertices areenumerated from 1 to n from to in the decreasing order. The vertex rs is the vertex number k + 1. Example 1.8.
Choosing rs = , we have the following picture:
01 11 43 75 32 21 10 ( L ) b b bb bb b where we have colored in pink the triangles at the left of ( L ) and in blue those at the right of ( L ). Notethat the lowest triangle can be viewed either at the left or at the right of ( L ). This is precisely thetriangulation T / (cf. Example 1.3) viewed inside the Farey tessellation.2. Matrices of negative and regular continued fractions
It is convenient to use 2 × , Z ) and allow the operations, such as multiplication, inverse,transposition; see [32]. Another reason which is particularly important for us is that matrices are more“perennial” than continued fractions. They continue to exist when continued fractions are not well-defined(because of potential zeros in the denominators) and enjoy similar properties.In this section, however, we still assume that the continued fractions are well-defined. Consider, as inSection 1, a rational number expanded into continued fractions: rs = J c , . . . , c k K = [ a , . . . , a m ] . The information about these expansions is contained in the matrices(2.1) M ( c , . . . , c n ) := c −
11 0 ! c −
11 0 ! · · · c n −
11 0 ! and(2.2) M + ( a , . . . , a m ) := a
11 0 ! a
11 0 ! · · · a m
11 0 ! . Both matrices are elements of SL(2 , Z ).The goal of this introductory section is to compare these two matrices and rewrite one from another.This, in particular, implies formula (1.2). The end of the section contains motivations for the sequel.2.1. The matrices of continued fractions.
The matrices (2.1) and (2.2) are known as the matricesof continued fractions, because one has the following statement whose proof is elementary.
Proposition 2.1.
One has M ( c , . . . , c k ) = r − r ′ s − s ′ ! , M + ( a , . . . , a m ) = r r ′′ s s ′′ ! , where rs = [ a , . . . , a m ] = J c , . . . , c k K , and where r ′ s ′ = J c , . . . , c k − K , and r ′′ s ′′ = [ a , . . . , a m − ] . Therefore, the matrices M + ( a , . . . , a m ) and M ( c , . . . , c k ) have the same first column, but they aredifferent. There exists a simple relationship between these matrices. Proposition 2.2.
One has: (2.3) M + ( a , . . . , a m ) = M ( c , . . . , c k ) R, where R = ! . Proof.
Formula (2.3) can be easily obtained using the results of the previous section. Indeed, in thetriangulation T r/s labeled as in (1.3), we see that r ′ s ′ , rs , r ′′ s ′′ label the vertices k, k + 1 , k + 2, respectively.This implies rs = r ′ s ′ + r ′′ s ′′ and hence (2.3). (cid:3) Alternatively and independently, the relation between the matrices M + ( a , . . . , a m ) and M ( c , . . . , c k )can be established by elementary matrix computations. This will be done in the next sections. AREY BOAT 11
Example 2.3.
Choosing, as in Example 1.3, the rational rs = , one obtains M + (1 , , ,
1) = ! , M (2 , ,
3) = − − ! . Note that these matrices have different traces and therefore cannot be conjugacy equivalent.2.2.
Matrices M + ( a , . . . , a m ) and M ( c , . . . , c k ) in terms of the generators. It will be useful tohave the expressions of M + ( a , . . . , a m ) and M ( c , . . . , c k ) in terms of the generators of SL(2 , Z ). Thefollowing formulas are standard and can be found in many sources. Proposition 2.4.
The matrices M + ( a , . . . , a m ) and M ( c , . . . , c k ) have the following decompositions M + ( a , . . . , a m ) = R a L a R a L a · · · R a m − L a m , (2.4) M ( c , . . . , c k ) = R c S R c S · · · R c k S, (2.5) where (2.6) R = ! , L = ! , S = −
11 0 ! . For the sake of completeness, we give here an elementary proof.
Proof.
Formula (2.4) is obtained from the elementary computation a i
11 0 ! a i +1
11 0 ! = a i a i +1 + 1 a i a i +1 ! = a i ! a i +1 ! = R a i L a i +1 . Formula (2.5) is obviously obtained from c i −
11 0 ! = R c i S. (cid:3) Converting the matrices.
The matrix M + ( a , . . . , a m ) with a i ≥ Proposition 2.5.
One has: (2.7) M + ( a , . . . , a m ) = − M (cid:0) a + 1 , , . . . , | {z } a − , a + 2 , , . . . , | {z } a − , . . . , a m − + 2 , , . . . , | {z } a m , , (cid:1) . Let us stress that (2.7) is equivalent to (2.3) under the assumption that we already know formula (1.2).However, our strategy is different, we use (2.7) to prove (1.2).We will need the following lemma.
Lemma 2.6.
One has R a = − M ( a + 1 , , and L a = − M (1 , , . . . , | {z } a , , . Proof.
With a direct computation one easily obtains M ( a + 1 , ,
1) = − R a . For the second formula weuse the following preliminary result that is easily obtained by induction −
11 0 ! a = a + 1 − aa − ( a − ! . Then a direct computation leads to M (1 , , . . . , | {z } a , ,
1) = − L a . Hence the lemma. (cid:3) Proof of Proposition 2.5.
Since M (1 , ,
1) = − Id, one gets from Lemma 2.6 R a i L a i +1 = − M ( a i + 1 , , . . . , | {z } a i +1 , , . Formula (2.7) then follows from (2.4) and the simple relation M (2 , , , a + 1) = − M ( a + 2).Proposition 2.5 is proved.Finally, we observe that the last three coefficients in (2.7) are (2 , , M (2 , ,
1) = − R . So that one gets(2.8) M + ( a , . . . , a m ) = M (cid:0) a + 1 , , . . . , | {z } a − , a + 2 , , . . . , | {z } a − , . . . , a m − + 2 , , . . . , | {z } a m − (cid:1) R. According to Proposition 2.1 the first column of the matrices from the right-hand-side and from theleft-hand-side gives the rational rs . Therefore, this establishes formula (1.2) and the relation (2.3).2.4. Converting the conjugacy classes in
PSL(2 , Z ) . We obtain it as a corollary of Proposition 2.5.
Corollary 2.7.
The matrix M + ( a , . . . , a m ) is conjugacy equivalent to the matrix M (cid:0) a + 2 , , . . . , | {z } a − , a + 2 , , . . . , | {z } a − , . . . , a m − + 2 , , . . . , | {z } a m − (cid:1) . Proof.
This statement immediately follows from (2.8) using conjugation by R . (cid:3) The integers(2.9) ( c , . . . , c k ) = (cid:0) a + 2 , , . . . , | {z } a − , a + 2 , , . . . , | {z } a − , . . . , a m − + 2 , , . . . , | {z } a m − (cid:1) appearing in the above formula were used to describe the conjugacy classes of PSL(2 , Z ); see [35, p.91]and provide interesting characteristics of the quadratic irrationalities. Example 2.8.
Let us go back to Example 2.3 that treats the case of the rational rs = . Applying (2.8),we get that M + (1 , , ,
1) is conjugacy equivalent to M (3 , ,
3) = − − ! . Appearance of the equation M ( c , . . . , c n ) = − Id . It turns out that, taking into account all thecoefficients ( c , . . . , c n ) of the triangulation T r/s (and not only ( c , . . . , c k ) as we did before), one obtainsthe negative of the identity matrix. The following statement can be found in [3]. Proposition 2.9.
One has M ( c , . . . , c n ) = − Id . Proof.
Rewrite M ( c , . . . , c n ) = M ( c , . . . , c k ) M (1) M ( c k +2 , . . . , c n − ) M (1) , then (2.8) together with Corollary 1.4 and Proposition 2.2 imply M ( c , . . . , c n ) = M + ( a , . . . , a m ) R − M (1) M + ( a m , . . . , a ) R − M (1)= M + ( a , . . . , a m ) S M + ( a m , . . . , a ) S, where S is as in (2.6). Since M + ( a m , . . . , a ) = M + ( a , . . . , a m ) t , we conclude using the fact that ASA t S = − Id for all A ∈ SL(2 , Z ). (cid:3) AREY BOAT 13
Every rational number rs thus corresponds to a solution of the equation M ( c , . . . , c n ) = − Id. Thisequation will be important in the sequel for two reasons.Firstly, the equation M ( c , . . . , c n ) = − Id makes sense and remains an interesting equation in general,when there is no particular rational number and the corresponding continued fraction. Expanding arational in a continued fraction rs = J c , . . . , c k K , we always assumed c i ≥
2. This assumption makes theexpansion unique. Allowing c i = 1 for some i , one faces two difficulties: the expansion is no more unique(there is an infinite number of them), and furthermore, the continued fraction may not be well-defined(the denominators may vanish). It turns out that considering the matrices M ( c , . . . , c k ) with c i ≥ , Z ) (and PSL(2 , Z )) in the form A = M ( c , . . . , c n ) for some positive integers c i . Therefore, it will be important to know the relationsleading to different presentations of the same element.2.6. The semigroup Γ . The matrices M + ( a , . . . , a m ) of regular continued fractions do not representarbitrary elements of PSL(2 , Z ). Definition 2.10.
The semigroup Γ ⊂ SL(2 , Z ) consists of the elements M + ( a , . . . , a m ) where a i arepositive integers.As mentioned in Proposition 2.4, Γ is generated by the matrices R and L . It consists of the matriceswith positive entries satisfying the following conditions:Γ = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) a ≥ b ≥ d > ,a ≥ c ≥ d > (cid:27) . The semigroup Γ is the main character of a wealth of different problems of number theory, dynamics,combinatorics, etc. It was studied by many authors from different viewpoints; see [2, 26, 5, 7] andreferences therein.This is one of the motivations for a systematic study of the matrices M ( c , . . . , c n ) which is one of themain subjects of this paper.3. Solving the equation M ( c , . . . , c n ) = ± IdIn this section we describe all positive integer solution of the two equations M ( c , . . . , c n ) = − Id , and M ( c , . . . , c n ) = Id , for the matrices (2.1). Recall that matrices M ( c , . . . , c n ) with c i ≥
1, satisfying M ( c , . . . , c n ) = − Id,arose from continued fractions, see Section 2.5. The equation M ( c , . . . , c n ) = Id is quite different butalso relevant.One motivation for considering solutions with arbitrary positive integers c i ≥ d -dissections” of n -gons. Another motivation is to extend most of the results and ideas of Section 1 fromcontinued fractions to arbitrary solutions of the equation M ( c , . . . , c n ) = ± Id. Our third motivation isrelated to a more general study (see Section 6) of decomposition of an arbitrary element A ∈ PSL(2 , Z ) inthe form A = M ( c , . . . , c n ). Solutions of the above equations describe relations in such a decomposition.Let us also mention that equation M ( c , . . . , c n ) = − Id considered over C defines an interestingalgebraic variety closely related to the classical moduli space M ,n of configurations of points in theprojective line. Therefore, positive integer solutions of this equation correspond to a class of rationalpoints of M ,n ; see [30]. We do not consider geometric applications in the present paper. Conway and Coxeter totally positive solutions.
A classical theorem of Conway and Cox-eter [11] describes a particular class of solutions of the equation(3.1) M ( c , . . . , c n ) = − Id . More importantly, this theorem relates this equation to combinatorics.The following notion is the most important ingredient of the theory.
Definition 3.1. (a) Given a triangulation of a convex n -gon by non-crossing diagonals, its quiddity is the(cyclically ordered) n -tuple of positive integers, ( c , . . . , c n ), counting the number of triangles adjacentto the vertices.(b) Given an n -tuple of positive integers, ( c , . . . , c n ), we consider the following sequence of rationalnumbers, or infinity: r i s i := J c , . . . , c i K , for 1 ≤ i ≤ n .For example, the coefficients c i of a negative continued fraction of a rational number rs is a part of thequiddity of the triangulation T r/s , and the rationals r i s i are its convergents; see Section 1.2. Of course,for continued fractions, the denominator of r i s i cannot vanish. Definition 3.2.
The class of solutions of (3.1) satisfying the condition(3.2) r i s i > , for all i ≤ n −
3, will be called totally positive .We will see in Section 3.3 that the above condition of total positivity is equivalent to the assumptionthat c + c + · · · + c n = 3 n − n -gon, via the notion of quiddity that uniquely determinesthe triangulation. Theorem 3.3 ([11]) . (i) The quiddity of a triangulated n -gon is a totally positive solution of (3.1).(ii) A totally positive solution of (3.1) is the quiddity of a triangulated n -gon. We do not dwell on the detailed proof of this classical result. For a simple complete proof of Theorem 3.3see [3, 20], and also [31]. The idea of the proof consists of three observations.1) An n -tuple of integers ( c , . . . , c n ) satisfying (3.1) must contain c i = 1 for some i . Otherwise, for anysequence of integers ( v i ) i ∈ Z satisfying the linear recurrence v i +1 = c i v i − v i − , with the initial conditions( v , v ) = (0 , v i +1 > v i . Therefore, the sequence ( v i ) i ∈ Z cannot be periodic. This contradictsthe equation M ( c , . . . , c n ) (cid:18) (cid:19) = ± (cid:18) (cid:19) .2) The total positivity condition (3.2) implies that, whenever c i = 1 for some i , the two neighbors c i − , c i +1 must be greater or equal to 2. Indeed, for two consecutive 1’s, if ( c i , c i +1 ) = (1 , v i +2 = v i +1 − v i = v i − v i − − v i = − v i − .3) The “local surgery” operation(3.3) ( c , . . . , c i − , , c i +1 , . . . , c n ) → ( c , . . . , c i − − , c i +1 − , . . . , c n )is then well-defined. It decreases n by 1, and does not change the matrix (2.1). AREY BOAT 15
Indeed, M ( c , . . . , c i − , , c i +1 , . . . , c n ) = M ( c , . . . , c i − − , c i +1 − , . . . , c n ) since c + 1 −
11 0 ! −
11 0 ! c ′ + 1 −
11 0 ! = cc ′ − − c ′ c − ! = c −
11 0 ! c ′ −
11 0 ! . One then proceeds by induction on n , the induction step consists of cutting an exterior triangle of agiven triangulation, that corresponds to the operation (3.3) on the quiddity. Example 3.4.
The sequence (2 , , , , , , , , , , , , , , ,
1) is a solution of (3.1) because it is thequiddity of the following triangulation of a hexadecagon:5 ❥❥❥❥❥ ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀❀ ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ ❚❚❚❚❚ ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ ⑧⑧⑧⑧✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭ ❄❄❄❄ ✴✴✴✴✴✴✴✴✴✴✴✴✴ ✎✎✎ ✴✴✴ ✬✬✬✬✬✬✬ ✬✬✬✬✬✬✬ ✴✴✴ ✎✎✎ ❄❄❄❄ ⑧⑧⑧⑧ ❚❚❚❚❚ ❥❥❥❥❥ ❞❞❞❞❞❞❞❞❞❞❞❞ Remark 3.5. a) The meaning of total positivity will be explained in Sections 4.1 and 5.3.b) The Conway and Coxeter theorem is initially formulated in terms of frieze patterns . This notion,due to Coxeter [12], became popular mainly because its relations to cluster algebras; see [8]. Friezepatterns also play an important role in such areas as quiver representations, differential geometry, discreteintegrable systems (for a survey; see [27]).3.2.
The complete set of solutions: d -dissections. It turns out that, to classify all the solutions(with no total positivity condition), it is natural to solve simultaneously the equations M ( c , . . . , c n ) = − Id and M ( c , . . . , c n ) = Id . This classification led to the following combinatorial notion.
Definition 3.6. (i) A 3 d - dissection is a partition of a convex n -gon into sub-polygons by means ofpairwise non-crossing diagonals, such that the number of vertices of every sub-polygon is a multiple of 3.(ii) The quiddity of a 3 d -dissection of an n -gon is defined, similarly to the case of a triangulation,as a cyclically ordered sequence ( c , . . . , c n ) of positive integers counting sub-polygons adjacent to everyvertex.The following theorem was proved in [31]. Theorem 3.7 ([31]) . (i) The quiddity of a d -dissection of an n -gon satisfies M ( c , . . . , c n ) = ± Id .(ii) Conversely, every solution of the equation M ( c , . . . , c n ) = ± Id with positive integers c i is thequiddity of a d -dissection of an n -gon. Similarly to Theorem 3.3, the proof uses induction on n . The idea is as follows. Besides the opera-tions (3.3), one needs another type of “local surgery” operations. These operations remove two consecutive c , . . . , c i − , c i , , , c i +3 , c i +4 , . . . , c n ) → ( c , . . . , c i − , c i + c i +3 − , c i +4 , . . . , c n ) . Such an operation decreases n by 3 and changes the sign of the matrix M ( c , . . . , c n ). Indeed, c ′ −
11 0 ! −
11 0 ! c ′′ −
11 0 ! = − c ′ − c ′′ − ! . Example 3.8.
Simple examples of 3 d -dissections different from triangulations are:1 ❖❖♦♦ ✎✎ ✴✴ ✴✴ ✎✎ ❥❥❥❥ ❚❚❚❚ ✠✠ ✻✻ ✻✻ ✠✠ ❚❚❚❚ ❥❥❥❥ M (1 , , , , , ,
1) = Id , M (1 , , , , , , , , ,
1) = − Id . Remark 3.9.
Theorem 3.7 does not imply a one-to-one correspondence between solutions of (0.2) and3 d -dissections. Moreover, such a correspondence does not exist. Indeed, the quiddity of a 3 d -dissectiondoes not characterize it. This means that different 3 d -dissections may correspond to the same quiddity.For instance, the following different 3 d -dissections of the octagon1 ☎☎ ✿✿ ✱✱✱✱✱ ✿✿ ☎☎✱✱✱✱✱ ☎☎ ✿✿ ❧❧❧❧❧❧❧ ✿✿ ☎☎❧❧❧❧❧❧❧ − Id and Id.
Corollary 3.10.
Given a d -dissection of an n -gon, its quiddity ( c , . . . , c n ) satisfies M ( c , . . . , c n ) = − Id if and only if the number of subpolygons with an even number of vertices is even. The total sum c + · · · + c n . An interesting characteristics of a solution is the total sum of thecoefficients c i .Theorem 3.7 implies that the value c + · · · + c n = 3 n − c i ’s is equal to 3 n − n -gon. Corollary 3.11.
Positive integer solutions of the equation M ( c , . . . , c n ) = ± Id always satisfy c + c + · · · + c n ≤ n − . The total sum can be expressed in terms of the subpolygons of the corresponding 3 d -dissection. Thefollowing statement is a combination of Corollary 2.3 and Proposition 3.1 of [31], we do not dwell on theproof here. AREY BOAT 17
Proposition 3.12.
The total sum of c i ’s in the quiddity of a d -dissection of an n -gon is c + c + · · · + c n = 3 n − X k ≤ [ n ] ( k − N k − , where N k is the number of k -gons in the d -dissection. It follows that the total sum of c i ’s can vary by multiples of 6, and the sign on the right-hand sideof M ( c , . . . , c n ) = ± Id alternates.
Corollary 3.13.
The solutions of the equation M ( c , . . . , c n ) = ± Id can be ranged by levels: (3.5) c + c + · · · + c n = 3 n − , ( − Id)= 3 n − , (Id)= 3 n − , ( − Id) . . . Walks on the Farey graph
In this section we show that every solution of the equation M ( c , . . . , c n ) = ± Id admits an embeddinginto the Farey tessellation. This is a generalization of the construction from Section 1.6.In particular, a totally positive solution corresponding to a triangulation of the n -gon, defines amonotonously decreasing walk from to . This is an n -cycle in the Farey graph that we refer toas a “Farey n -gon”. The Farey tessellation then induces a triangulation which coincides with the initialtriangulation.A more general solution corresponding to a 3 d -dissection of an N -gon defines (an oriented) walk alonga certain Farey n -gon, where n < N . Every such walk is an N -cycle, and we show that the quiddity ofthe 3 d -dissection of the N -gon can be recovered from the triangulation of the Farey n -gon.4.1. Solutions of M ( c , . . . , c n ) = − Id and n -cycles in the Farey graph. We use the followingcombinatorial data in the Farey graph.
Definition 4.1. (i) An n -cycle in the Farey graph is a sequence ( v i ) i ∈ Z of vertices (with the cyclic orderconvention v i + n = v i ), such that v i − and v i are connected by an edge for all i .(ii) We call a Farey n -gon every n -cycle in the Farey graph such that v = 10 , v n − = 01 , and v i − > v i , for all i = 1 , . . . , n − Example 4.2.
The sequence (cid:8) , , , , , , , , , , , , , , , (cid:9) is a Farey hexadecagon: b bb bb b bb b b bb b b b b
01 11 54 97 1021324317136449111851581214736302313102217
Let ( c , . . . , c n ) be a set of positive integers such that M ( c , . . . , c n ) = − Id. Our next goal is to definethe corresponding n -cycle in the Farey graph.We define a sequence of n vertices in the Farey graph (cid:16) r s , . . . , r n − s n − (cid:17) that starts with and endswith with the following recurrence relations:(4.1) (cid:26) r i := c i r i − − r i − s i := c i s i − − s i − . and the initial conditions r − s − = − , r s = . We get a sequence of the form(4.2) (cid:18) r s , . . . , r n − s n − (cid:19) = (cid:18) , c , c c − c , c c c − c − c c c − , . . . , (cid:19) ;The fact that r n − s n − = follows from the relation M ( c , . . . , c n ) = − Id. Indeed, inductively one obtains M ( c , . . . , c i ) = r i − r i − s i − s i − ! , where r i s i = J c , . . . , c i K . is the sequence of convergents of the negative continued fraction J c , . . . , c n K . In particular, − Id = M ( c , . . . , c n ) = r n − r n − s n − s n − ! , so r n − = 0 and s n − = 1, and furthermore r n = − − r and s n = 0 = s . This implies that the sequences defined by (4.1) are n -antiperiodic, and one obtains asequence (cid:16) r i s i (cid:17) i ∈ Z , such that r i + n s i + n = − r i − s i . Proposition 4.3. (i) If ( c , . . . , c n ) is a positive solution of M ( c , . . . , c n ) = − Id , then the sequence (4.2) is an n -cycle in the Farey graph.(ii) If ( c , . . . , c n ) is a totally positive solution, then the sequence (4.2) is a Farey n -gon. AREY BOAT 19
Proof.
Part (i). For every pair of sequences, ( r i ) and ( s i ), satisfying the linear recurrence (4.1) oneobtains constant 2 × r i r i − s i s i − ! = det − r i − r i − − s i − s i − ! = det r i − r i − s i − s i − ! = . . . = det r r − s s − ! = − . Therefore, r i s i and r i − s i − are connected by an edge.Part (ii). The positivity condition (3.2) and the relation r i s i − − r i − s i = − r i s i < r i − s i − .Hence the result. (cid:3) It turns out that every Farey n -gon gives rise to a totally positive solution, so that we can formulatethe following statement. For details; see [29, Proposition 2.2.1]. Corollary 4.4.
Totally positive solutions of the equation M ( c , . . . , c n ) = − Id are in one-to-one corre-spondence with Farey n -gons. Remark 4.5.
The solution ( c , . . . , c n ) can be recovered from the n -gon using the notion of index of aFarey sequence (this notion was defined and studied in [18]). Given an n -gon (cid:16) r s , r s , . . . , r n − s n − (cid:17) in theFarey graph, its index is the n -tuple of integers c i := r i − + r i +1 r i = s i − + s i +1 s i . In our terms, the index is nothing else than the quiddity of the triangulation.4.2.
Farey n -gons and triangulations. Classical properties on the Farey graph imply the followingstatement whose proof can be found in [29].
Proposition 4.6.
Every Farey n -gon is triangulated in the Farey tessellation. In other words, the full subgraph of the Farey graph, containing the vertices of a Farey n -gon forms atriangulation of the n -gon. Example 4.2 gives an illustration of this statement.Therefore from a totally positive solution of M ( c , . . . , c n ) = − Id, one obtains two triangulations: onegiven by Conway and Coxeter’s correspondence (see Theorem 3.3, Part (ii)), and the other one given byFarey n -gons, see Corollary 4.4. Theorem 4.7 ([29], Theorem 1) . The Conway and Coxeter triangulation coincides with the Farey tri-angulation.
We do not dwell on the proof here (see [29, Section 2.2]).The rational labels on the vertices of the triangulation can be recovered directly from the triangulationby the following combinatorial algorithm. Note that this is the same algorithm as in Section 1.4, exceptfor step (1), and applied to arbitrary triangulations.(1) In the triangulated n -gon, label the vertex number 1 by and the vertex number n by .(2) Label all the vertices of the n -gon according to the rule: Whenever two vertices of the sametriangle have been assigned the rationals r ′ s ′ and r ′′ s ′′ , then the third vertex receives the label r ′ s ′ ⊕ r ′′ s ′′ := r ′ + r ′′ s ′ + s ′′ . Example 4.8.
Applying the above rule for the rational labelling on the hexadecagon of Example 3.4,one obtains ✐✐✐✐✐✐✐ ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ ❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂❂ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ ❯❯❯❯❯❯❯ ❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁❁ ⑥⑥⑥⑥✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭✭ ❆❆❆❆❆ ✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵✵ ✍✍✍ ✵✵✵ ✬✬✬✬✬✬✬✬✬ ✬✬✬✬✬✬✬✬✬ ✵✵✵ ✍✍✍ ❆❆❆❆ ⑥⑥⑥⑥⑥ ❯❯❯❯❯❯❯ ✐✐✐✐✐✐✐ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ which coincides with the Farey hexadecagon of Example 4.2. Note that the triangulation induced fromthe Farey tessellation coincides with the initial triangulation in Example 3.4.Consider a cyclic permutation of ( c , . . . , c n ). This gives another solution of M ( c , . . . , c n ) = − Id.Clearly the corresponding triangulations given by Theorem 3.3, Part (ii) are related by a cyclic permu-tation of the vertices, i.e. a rotation.
Proposition 4.9.
The Farey n -gons corresponding to a cyclic permutation of ( c , . . . , c n ) are related bya cyclic permutation modulo the action of SL(2 , Z ) by linear-fractional transformations. This statement is proved in [29, Proposition 2.2.1].
Example 4.10.
Let n = 6, and consider the totally positive solutions (3 , , , , ,
1) and (1 , , , , , (cid:18) , , , , , (cid:19) and (cid:18) , , , , , (cid:19) and to the triangulated hexagons ☛☛ ◆◆◆◆◆◆◆◆ ✸✸ ✸✸ ☛☛ ♣♣♣♣♣♣♣♣ and ☛☛ ✸✸♣♣♣♣♣♣♣♣ ✸✸ ◆◆◆◆◆◆◆◆ ☛☛
13 12 respectively. One checks that the first Farey hexagon is obtained from the second one by the action ofthe matrix −
11 0 ! . d -dissections and walks on the Farey tessellation. Construction (4.1) can be applied with anarbitrary positive solution of the equation M ( c , . . . , c N ) = ± Id. This leads again to an N -cycle startingat and ending at .In the sequence of vertices defining the cycle a vertex may appear several times and it will be importantto distinguish rs and − r − s . In other words, we will consider the twofold covering of the projective line over Q . AREY BOAT 21
Definition 4.11.
Given a Farey n -gon, (cid:16) , r s , . . . , r n − s n − , (cid:17) and an integer N ≥ n , an N -periodic (orantiperiodic) sequence of its vertices, (cid:0) r ij s ij (cid:1) j ∈ Z , is called(i) a walk on the n -gon if r ij s ij and r ij +1 s ij +1 are connected by an edge, for all j ;(ii) a positive walk if it is a walk and r i j s i j +1 − s i j r i j +1 >
0, for all j .In other words, fixing an orientation on the Farey n -gon, a positive walk has the same orientation. Example 4.12. a) A Farey n -gon itself is an n -antiperiodic positive walk. The antiperiodicity is due tothe fact that after arrival at r n s n = , one has to continue with − in order to keep r n s n +1 − s n r n +1 = 1.Let us give simple concrete examples of positive walks.b) Consider the Farey quadrilateral (cid:0) , , , (cid:1) , the 7-periodic walk (cid:16) , , , − , − − , − − , − (cid:17) ispositive. The quadrilateral and its Farey triangulation and the walk are represented by the diagrams: ❄❄❄❄❄⑧⑧⑧⑧⑧
11 0112 ⑧⑧⑧⑧⑧❄❄❄❄❄ ❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ − ❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ / / _ _ ❄ ❄ ❄ − − − _ _ ❄ ❄ ⑧⑧⑧⑧⑧❄❄❄❄❄ − − ? ? ⑧⑧⑧ ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ where the dashed arrow indicates a change of signs in the sequence so that the next step of the walk isdrawn in the copy of the n -gon with opposite signs.The following 10-antiperiodic walk (cid:16) , , , , − − , − − , − , , , (cid:17) along the same quadrilateral isalso positive and is represented by ❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ − ❄❄❄⑧⑧⑧ ❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ o o ❴❴❴❴❴ − − − _ _ ❄ ❄ / / _ _ ❄ ❄ ❄ ? ? ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ − − ? ? ⑧⑧⑧ ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ⑧⑧⑧⑧⑧❄❄❄❄❄ c) Consider the Farey hexagon (cid:0) , , , , , (cid:1) : ☛☛ ✸✸♣♣♣♣♣♣♣♣ ✸✸ ◆◆◆◆◆◆◆◆ ☛☛
13 12
The 9-periodic walk (cid:16) , , , , , − − , − − , − − , − (cid:17) is positive and is represented by the diagram ☛☛ / / ✸✸♣♣♣♣♣♣♣♣ (cid:15) (cid:15) − ☛☛ − − (cid:25) (cid:25) ✸✸♣♣♣♣♣♣♣ ✸✸✸✸ Y Y ✸✸ ◆◆◆◆◆◆◆◆✸✸ ♣♣♣♣ ☛☛ − ✸✸ f f ◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆ ♣♣♣♣ − − (cid:5) (cid:5) ☛☛☛☛ o o − − − −
22 SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO
Recall (cf. Section 4.1) that every solution of the equation M ( c , . . . , c N ) = ± Id defines an N -(anti)periodic positive walk on some Farey n -gon, where n ≤ N . The following theorem is nthe mainresult of this section. Theorem 4.13. (i) Every N -(anti)periodic positive walk on a Farey n -gon corresponds to a solution ofthe equation M ( c , . . . , c N ) = ± Id .(ii) Conversely, every solution of M ( c , . . . , c N ) = ± Id can be obtained from an N -(anti)periodic walkon Farey n -gons with n ≤ N .Proof. Part (i). Consider an N -(anti)periodic positive walk (cid:16) r i s i (cid:17) ≤ i ≤ N . Since for every i we have(4.3) det r i r i +1 s i s i +1 ! = r i s i +1 − r i +1 s i = 1 , both, the numerator and the denominator must satisfy a linear recurrence V i +1 = c i V i − V i − , with some N -periodic sequence ( c i ) i ∈ Z . The monodromy of this equation is the matrix M ( c , . . . , c N ).Since ( r i ) i ∈ Z and ( s i ) i ∈ Z are two (anti)periodic solutions, this monodromy is equal to ± Id.Part (ii). Given a solution of the equation M ( c , . . . , c N ) = ± Id, applying the construction (4.2), oneobtains a sequence of rationals (cid:0) r i s i (cid:1) i ∈ Z , which is periodic or antiperiodic depending on the sign in theright-hand-side of the equation and satisfies (4.3). To prove that this sequence is indeed a walk on aFarey n -gon, one needs to show that every pair of neighbors r i s i and r j s j (i.e., such points that there is noother point in the sequence in the interval (cid:0) r i s i , r j s j (cid:1) ) is connected by an edge in the Farey graph. Supposethat r i s i and r j s j are not connected. Assume that r i s i < r j s j . Then either r i +1 s i +1 > r j s j , or r i − s i − > r j s j , and both ofthese points must be connected to r i s i . Similarly, either r j +1 s j +1 < r i s j , or r j − s j − < r i s i , and both of these pointsmust be connected to r j s j . This means that there are crossing edges (geodesics in the Farey tessellation),which is a contradiction. (cid:3) Theorem 4.13 establishes a one-to-one correspondence between positive N -(anti)periodic walks on theFarey tessellation and solutions of M ( c , . . . , c N ) = ± Id. Theorem 4.13 together with Theorem 3.7 thenimply the following.
Corollary 4.14.
The sequence ( c , . . . , c N ) corresponding to an N -(anti)periodic positive walk is a quid-dity of a d -dissection of an N -gon. Example 4.15.
Let us continue Example 4.12.The 7-periodic walk (cid:16) , , , − , − − , − − , − (cid:17) generates the 7-periodic sequence ( c i ) i ∈ Z with theperiod (1 , , , , , , (cid:16) , , , , − − , − − , − , , , (cid:17) generates the 10-periodic sequence ( c i ) i ∈ Z with the period (1 , , , , , , , , , d -dissections1 ❖❖♦♦ ✎✎ ✴✴ ✴✴ ✎✎ ❥❥❥❥ ❚❚❚❚ ✠✠ ✻✻ ✻✻ ✠✠ ❚❚❚❚ ❥❥❥❥ AREY BOAT 23
The 9-periodic walk (cid:16) , , , , , − − , − − , − − , − (cid:17) generates the quiddity of the following 3 d -dissection of a nonagon: 1 ❥❥❥❥ ❚❚❚❚ ✠✠ ✻✻ ✻✻✻✻✻✻ ✠✠✠✠✠✠ ❍❍❍❍ ✈✈✈✈ n -gon, the “invisible hand” draws a 3 d -dissection ofan N -gon with N > n .4.4.
The quiddity of a d -dissection from a Farey walk. Given an n -gon, (cid:16) , r s , . . . , r n − s n − , (cid:17) , andan N -(anti)periodic walk on it (cid:0) r ij s ij (cid:1) j ∈ Z , it is natural to ask, how to recover the sequence ( c , . . . , c N )which is a solution of the equation M ( c , . . . , c N ) = ± Id. The answer is as follows.The integer c i j counts the number of triangles in the n -gon that lie on the positive side of the walk, r ij +1 s ij +1 ②②②②②②② < < ②②②②②②② r ij s ij b b ❊❊❊❊❊❊❊ ❊❊❊❊❊❊❊✱✱✱✱✱✱ ❳❳❳❳❳❳❳❳❳❳❳ ❢❢❢❢❢❢❢❢❢❢❢✒✒✒✒✒✒ ... r ij − s ij − with respect to the orientation of the hyperbolic plane. This is a quiddity of a 3 d -dissection of an N -gon,as follows from Theorem 3.7. 5. PPP: Ptolemy, Pl¨ucker and Pfaff
In this section we prove that every solution ( c , . . . , c n ) of the equation M ( c , . . . , c n ) = ± Id, with c i positive integers, defines a certain labeling of the diagonals of a convex n -gon: x : V × V → Z , where V is the set of vertices of the n -gon, usually identified with { , . . . , n } . Moreover, the set ofintegers x i,j , satisfies the Ptolemy-Pl¨ucker relations. In this sense, the results discussed in Section 1.5 arestill valid in the case where no continued fraction is defined. Note that the totally positive solutions arein a one-to-one correspondence with the labelings where all x i,j are positive integers. For an arbitrarysolution, one can only guarantee that the shortest diagonals are labeled by positive integers.This section contains the proofs of the main results. Our main tool is the well-known polynomialcalled (Euler’s) continuant. This is the determinant of a tridiagonal matrix, it gives an explicit formulafor the entries of the matrices M ( c , . . . , c n ). The Ptolemy-Pl¨ucker relations are deduced from the Euleridentity for the continuants.We conclude the section with the similar “Pfaffian formulas” for the trace tr( M ( c , . . . , c n )) recentlyobtained in [9]. The proof is more technical and we do not dwell on it. Continuant = “continued fraction determinant”. The material of this subsection is classical.Let us think of ( c , . . . , c n ) as formal commuting variables, and consider the negative continued fraction:(5.1) r n s n = J c , . . . , c n K , here and below n ≥
1. Then both the numerator and the denominator are certain polynomials in c i . Itturns out that these polynomials are basically the same. Definition 5.1.
The tridiagonal determinant K n ( c , . . . , c n ) := det c c . . . . . . . . . c n − c n is called the continuant . We also set for convenience K := 1 and K − := 0.The following statement is commonly known. The proof is elementary and we give it for the sake ofcompleteness. Proposition 5.2.
The numerator and the denominator of (5.1) are given by the continuants (5.2) ( r n = K n ( c , . . . , c n ) ,s n = K n − ( c , . . . , c n ) . Proof.
Formula (5.2) follows from the recurrence relation(5.3) V i +1 − c i +1 V i + V i − = 0 , with (known) coefficients ( c i ) i ∈ Z and (indeterminate) sequence ( V i ) i ∈ Z . Let r i s i = J c , . . . , c i K be a conver-gent of the continued fraction (5.1), then both sequences, ( r i ) i ≥ and ( s i ) i ≥ satisfy (5.3), with the initialconditions ( r , r ) = ( c , c c −
1) and ( s , s ) = (1 , c ). Indeed, this is equivalent to Proposition 2.1.On the other hand, the continuants satisfy(5.4) K i ( c , . . . , c i ) = c i K i − ( c , . . . , c i − ) − K i − ( c , . . . , c i − ) . Hence the result. (cid:3)
As a consequence of (5.2), we obtain the following formula for the entries of the matrix M ( c , . . . , c n ):(5.5) M ( c , . . . , c n ) = K n ( c , . . . , c n ) − K n − ( c , . . . , c n − ) K n − ( c , . . . , c n ) − K n − ( c , . . . , c n − ) ! . Indeed, M ( c , . . . , c n ) is the matrix of convergents, cf. Section 2.1.5.2. The Euler identity for continuants.
The polynomials K n ( c , . . . , c n ) were studied by Euler whoproved the following identity (see, e.g., [17]). Theorem 5.3 (Euler) . For ≤ i ≤ j < k ≤ ℓ ≤ n , one has (5.6) K k − i ( c i , . . . , c k − ) K ℓ − j ( c j +1 , . . . , c ℓ ) = K j − i ( c i , . . . , c j − ) K ℓ − i ( c i +1 , . . . , c ℓ ) + K ℓ − i +1 ( c i , . . . , c ℓ ) K k − j − ( c j +1 , . . . , c k − ) . We give here an elegant proof due to A. Ustinov [34] that makes use of the Pfaffian of a skew-symmetricmatrix.
AREY BOAT 25
Proof.
Using the notation x i − ,j +1 := K j − i +1 ( c i , . . . , c j ) , for i < j , consider the 4 × x i +1 ,j x i +1 ,k x i +1 ,ℓ − x i +1 ,j x j,k x j,ℓ − x i +1 ,k − x j,k x k,ℓ − x i +1 ,ℓ − x j,ℓ − x k,ℓ . It readily follows from the recurrence relation (5.4), that the matrix Ω has rank 2. Hencedet(Ω) = ( x i +1 ,j x k,ℓ + x i +1 ,ℓ x j,k − x i +1 ,k x j,ℓ ) = 0 , which is precisely (5.6). (cid:3) Remark 5.4. a) Formula (5.2) allows one to work with continued fractions with c i assigned to concretenumbers (integers, real, complex, etc.), even when the “naive” expression (5.1) is not well-defined. Thismay happen when some denominators vanish, for instance if several consecutive coefficients c i , c i +1 , . . . are equal to 1.b) Replacing the negative continued fraction by a regular one: r n s n = [ a , . . . , a n ], where n can be evenor odd, formula (5.2) is replaced by a similar formula with the only difference that the continuant isreplaced by the determinant K + n ( a , . . . , a n ) := det a − a . . . . . . . . . − a n − − a n , also known under the name of continuant.c) The continuants enjoy many remarkable properties (some of which are listed in [17, 4, 9]). Theywere already known to Euler who thoroughly studied the polynomials K n and, in particular, estab-lished Ptolemy-type identities for them. In a sense, the continuants establish a relationship between thecontinued fractions and projective geometry; see [30] and references therein.d) Let us mention that equation (5.3) is called the discrete Sturm-Liouville, Hill, or Schr¨odingerequation. It plays an important role in many areas of algebra, analysis and mathematical physics. Whenthe sequence of coefficients is periodic: c i + n = c i , for all i , there is a notion of monodromy matrix of (5.3),which is nothing else than the matrix M ( c , . . . , c n ).5.3. Ptolemy-Pl¨ucker relations.
We are ready to explain the connection between solutions of theequation M ( c , . . . , c n ) = ± Id and the Ptolemy-Pl¨ucker relations.Let x i,j , where i, j ∈ { , . . . , n } , be a set of n formal (commuting) variables. In order to have a clearcombinatorial picture, and following [15], we will always think of an n -gon with the vertices cyclicallyordered by { , . . . , n } , and the diagonals ( i, j ) labeled by x i,j .We call the Ptolemy-Pl¨ucker relations the following system of equations(5.7) x i,j x k,ℓ = x i,k x j,ℓ + x i,ℓ x k,j , i ≤ k ≤ j ≤ ℓ,x i,i = 0 ,x i,i +1 = 1 . We will distinguish two special cases, where the set of variables x i,j is either symmetric, or skew-symmetric: x i,j = x j,i , or x i,j = − x j,i . The following statement arose as an attempt to interpret some of the results of [12] (see also [30]).
Theorem 5.5.
Let ( c , . . . , c n ) be positive integers. The system (5.7) together with the symmetry condi-tion x i,j = x j,i has a unique solution such that x i − ,i +1 = c i if and only if one has M ( c , . . . , c n ) = − Id .Proof. We use the following lemma.
Lemma 5.6. If ( x i,j ) satisfies the Ptolemy-Pl¨ucker relations (5.7) then for all i ≤ j one has (5.8) x i − ,j +1 = det c i c i +1 . . . . . . . . . c j − c j = K j − i +1 ( c i , . . . , c j ) . where c i = x i − ,i +1 .Proof. We proceed by induction on j .The induction base is x i − ,i +1 = c i = K ( c i ) by definition, and the following calculation of x i − ,i +2 . i − c i ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ●●●●●●●●●●● i + 2 c i +1 ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ✇✇✇✇✇✇✇✇✇✇ x i − ,i +2 i i + 1 The Ptolemy-Pl¨ucker relation reads c i c i +1 = x i − ,i +2 + 1, hence x i − ,i +2 = K ( c i , c i +1 ).The induction step consists of expanding the determinant of (5.8) with respect to the last column andcompare with the Ptolemy-Pl¨ucker relation given by the diagram i − x i − ,j ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ x i − ,j − ●●●●●●●●●● j + 1 c j ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ✇✇✇✇✇✇✇✇✇✇✇ x i − ,j +1 j − j . Both relations are equivalent to (5.4). Hence the lemma. (cid:3)
Let us show that the Ptolemy-Pl¨ucker relations imply M ( c , . . . , c n ) = − Id. Applying Lemma 5.6 tothe case | j − i | = n −
1, and using the cyclic numeration of the vertices of the n -gon, we get x i,i − = K n ( x i , . . . , x i + n − ) = 1 , provided x i,j = x j,i . Then, again implying (5.7), one readily gets K n +1 ( x i , . . . , x i + n ) = 0 , K n +2 ( x i , . . . , x i + n +1 ) = − . We conclude by (5.5), that M ( c , . . . , c n ) = − Id.Conversely, assume that M ( c , . . . , c n ) = − Id. Starting from x i,i = 0, x i,i +1 = 1 and then labeling thediagonals of the n -gon using (5.8) one obtains a solution of (5.7) using the Euler identities (5.3) for thecontinuants. (cid:3) AREY BOAT 27
A similar computation (that we omit) allows one to prove the following skew-symmetric counterpartof Theorem 5.5.
Theorem 5.7.
Let ( c , . . . , c n ) be positive integers. The system (5.7) together with the skew-symmetrycondition x i,j = − x j,i has a unique solution such that x i − ,i +1 = c i if and only if M ( c , . . . , c n ) = Id . Remark 5.8.
Let us mention that the coordinates x i,j satisfying (5.7) together with the skew-symmetrycondition can be identified with the Pl¨ucker coordinates of the Grassmannian G ,n of 2-dimensionalsubspaces in the n -dimensional vector space. The coordinate ring of G ,n is one of the basic examplesof cluster algebras of Fomin and Zelevinsky [14] (for details; see [15]). A description of the relationshipbetween Coxeter’s frieze patterns and cluster algebras can be found in [27].5.4. Relation to d -dissections. So far in this section we considered formal variables x i,j . Assign-ing concrete integral values to these variables, one has to deal with integer solutions of the equation M ( c , . . . , c n ) = ± Id. In particular, Theorems 3.7, 5.5 and 5.7 imply the following relation to 3 d -dissections (see Section 3.2). Corollary 5.9.
Given an n -tuple of positive integers ( c , . . . , c n ) , start labeling the diagonals of an n -gonby x i,i = 0 , x i,i +1 = 1 , x i − ,i +1 = c i , for all ≤ i ≤ n , and then continue using the Ptolemy-Pl¨ucker relations. This procedure is consistent,and there exists a set of integers x i,j and satisfying (5.7), if and only if ( c , . . . , c n ) is a quiddity of a d -dissection. Traces and Pfaffians.
Let us give one more determinant formula. We are interested in calculatingthe trace of the matrix M ( c , . . . , c n ). It follows from (5.5) that this trace is equal to the difference oftwo continuants: tr( M ( c , . . . , c n )) = K n ( c , . . . , c n ) − K n − ( c , . . . , c n − ) . It turns out that the square of this polynomial is equal to the determinant of a 2 n × n matrix. Theorem 5.10 ([9, 10]) . The trace of the matrix M ( c , . . . , c n ) is equal to the square root of the deter-minant of the following skew-symmetric n × n matrix (5.9) det c c . . . . . . . . .. . . . . . − c n − c − − . . . . . .. . . . . . − − − c n − = (tr M ( c , . . . , c n )) . In other words, tr( M ( c , . . . , c n )) is the Pfaffian of the matrix on the left-hand-side of (5.9). Werefer to [9] for a proof of this result. Let us mention that formula (5.9) reflects a relation to symplecticgeometry; see [10]. More precisely, the 2 n × n matrix in (5.9) appears as the Gram matrix of thesymplectic form in the standard symplectic space evaluated on a Lagrangian configuration. This relationdeserves further investigation. Example 5.11.
In the case n = 3 one can easily check directly thatdet c c − c − c − − − c − − − c − = ( c c c − c − c − c ) , which is nothing other than the square of the trace of M ( c , c , c ). Remark 5.12.
If one wants to check (5.9) with the computer and forgets to put the minus sign, here iswhat one will obtain for n ≥ c c . . . . . . . . .. . . . . .
11 1 c n c . . . . . .. . . . . . c n = ( − n (cid:0) (tr M ( c , . . . , c n )) − (cid:1) . Note that the expression in the right-hand-side of (5.10) is the discriminant of the characteristic polyno-mial of M ( c , . . . , c n ). It will appear again in Section 6.1.6. Minimal presentation of
PSL(2 , Z )The group SL(2 , Z ) (and thus PSL(2 , Z )) is generated by two elements, and a standard choice ofgenerators is either { S, R } , { S, U } , or { U, R } , where S = −
11 0 ! , R = ! , U = −
11 0 ! . A natural question is how to make such a presentation canonical.It is a simple and well-known fact that every element A ∈ PSL(2 , Z ) can be presented in the form A = M ( c , . . . , c k ) where c i are positive integers. The above question is equivalent to the existence ofa canonical presentation in this form. This question was considered and answered (modulo conjugationof A ) in [22, 35, 24]. The coefficients ( c , . . . , c k ) are obtained as the (minimal) period of the negativecontinued fraction of a fixed point of A . This fixed point is a quadratic irrationality.We show the existence and uniqueness of the “minimal presentation”, A = M ( c , . . . , c k ), with c i ≥ k is the smallest possible. The coefficients ( c , . . . , c k ) of this presentation are calculated via expansionof a rational number (the quotient of largest coefficients of A ). This statement looks quite surprisingsince it recovers the period of a quadratic irrationality from a continued fraction of a rational. AREY BOAT 29
Parametrizing the conjugacy classes in
PSL(2 , Z ) . Let us outline the history of the problemdiscussed in this section. The matrices M ( c , . . . , c k ), with c i ≥ i , were used to parametrizeconjugacy classes of hyperbolic elements of PSL(2 , Z ) (recall that A ∈ PSL(2 , Z ) is hyperbolic if | tr A | ≥ RP , which is identified with R ∪ {∞} by choosing an affine co-ordinate x . The action of PSL(2 , Z ) on RP is given by linear-fractional transformations, viz A = (cid:18) a bc d (cid:19) : x → ax + bcx + d . When A is hyperbolic, it has two fixed points x ± ∈ RP : x ± = a − d ± p ( a + d ) − c . Note that the expression (tr A ) − A ∈ PSL(2 , Z )with tr A >
0, the point x + has the property that, for all x = x − , A m ( x ) tends to x + , when m → ∞ .The point x + is thus called the attractive fixed point of A .Since x ± are quadratic irrationals, the corresponding continued fractions are periodic (starting fromsome place) by Lagrange’s theorem; see, e.g., [32, 6]. Consider the negative continued fraction expansionof the attractive fixed point: x + = J c , . . . , c ℓ , c ℓ +1 , . . . , c ℓ + k K , where ( c ℓ +1 , . . . , c ℓ + k ) is the minimal period of the continued fraction.The statement explained in [35, pp.90–92] can be formulated as follows. Proposition 6.1.
Every hyperbolic element A ∈ PSL(2 , Z ) is conjugate to M ( c ℓ +1 , . . . , c ℓ + k ) , and the k -tuple ( c ℓ +1 , . . . , c ℓ + k ) , defined modulo cyclic permutations, characterises the conjugacy class of A uniquely. We refer to [22, 35] and [24] for a detailed and very clear treatment of this statement and its applica-tions.
Example 6.2.
Consider the matrix A = (cid:18)
10 33 1 (cid:19) , whose attractive fixed point is x + = √ . It’scontinued fraction expansion reads x + = [3 ,
3] = q , , , y . According to Proposition 6.1, the matrix A must be conjugate to M (2 , ,
5) = (cid:18) − − (cid:19) , and, indeed, one checks that A = M (4) M (2 , , M (4) − . Proposition 6.1 and its impact for number theory, see, e.g., [13] and references therein, is the mainmotivation for us to study minimal presentations of elements of PSL(2 , Z ). When representing a matrix,the condition c i ≥
2, for all i , cannot always be satisfied (many interesting matrices need 1’s at the endsof their minimal presentations).6.2. Minimal presentation.
Every element A ∈ PSL(2 , Z ) can be written in the form A = M ( c , . . . , c k ),where c i ≥
1. We are interested in the shortest presentations of this form. It turns out that the coeffi-cients c i can be recovered from the coefficients of A , without expansions of quadratic irrationals. Theorem 6.3. (i) The presentation A = M ( c , . . . , c k ) with positive integer coefficients c i is unique,provided k is the smallest possible.(ii) If A = (cid:18) a − bc − d (cid:19) where a, b, c, d > and a > b , then the coefficients ( c , . . . , c k ) are those of thecontinued fraction ac = J c , . . . , c k K . We also have the following “minimality criterion”.
Proposition 6.4.
If an element A of PSL(2 , Z ) is written in the form A = M ( c , . . . , c k ) , then this isthe minimal presentation of A , if and only if c i ≥ , except perhaps for the ends of the sequence, i.e., for c , or c , c and c k , or c k − , c k . Before giving the proof of Theorem 6.3 and Proposition 6.4, let us consider several examples andcorollaries.Part (ii) of Theorem 6.3 covers all different cases of elements of PSL(2 , Z ), modulo multiplicationby R and S from the right and from the left. For instance, the following statement treats the case of allmatrices with positive coefficients. Corollary 6.5.
Let A = (cid:18) a bc d (cid:19) be an element of PSL(2 , Z ) with a, b, c, d > , one has the followingtwo cases:(i) if a > b , then the minimal presentation of A is (6.1) A = M ( c , . . . , c k , , , , where ac = J c , . . . , c k K ; and the conjugacy class of A is parametrized by ( c + 1 , c , . . . , c k ) ;(ii) if a < b , then the minimal presentation of A is (6.2) A = M ( c , . . . , c k − , c k + 1 , , where bd = J c , . . . , c k K ; and the conjugacy class of A is parametrized by ( c + c k , c , . . . , c k − ) .Proof. Part (i). After multiplication from the right by R − , the matrix AR − satisfies the conditions ofPart (ii) of Theorem 6.3. One then uses that R − = M (1 , , ,
1) and M (2 , , ,
1) = Id (up to a sign,i.e., in PSL(2 , Z )). Hence (6.1). Next, one has RAR − = M (2 , , , c , . . . , c k ) = M ( c + 1 , c , . . . , c k ) . Part (ii). The matrix A becomes as in Part (ii) of Theorem 6.3, when multiplied from the right by S = M (1 , , , , R a = M ( a + 1 , , R c k AR − c k = M ( c k + 1 , , , c , . . . , c k − ) = M ( c + c k , c , . . . , c k − ) . Hence the result. (cid:3)
Remark 6.6.
Comparing (6.1) and (6.2) to somewhat similar known formulas in terms of the positivecontinued fractions (see [23], Theorem 7.14), we observe that they are quite different. Indeed, formu-las (6.1) and (6.2) use the “dominant” (largest) coefficients of A , while the formulas in [23] use thesmallest coefficients.Rewriting (6.1) and (6.2) in terms of the standard generators, and using Proposition 2.4, we have thefollowing decomposition. Corollary 6.7.
Let A = (cid:18) a bc d (cid:19) be an element of PSL(2 , Z ) with a, b, c, d > . Its expression in termsof the standard generators is:(i) if a > b , then A = R c S R c S · · · SR c k SR, = R a ( U R ) a · · · R a m − ( U R ) a m where ac = [ a , . . . , a m ] = J c , . . . , c k K ;(ii) if a < b , then A = R c S R c S · · · SR c k , = R a ( U R ) a · · · R a m − ( U R ) a m − R where bd = [ a , . . . , a m ] = J c , . . . , c k K . AREY BOAT 31
Together with Proposition 2.4, Theorem 3.7 implies an explicit description of relations in the groupPSL(2 , Z ). Every element A ∈ PSL(2 , Z ) can be written in terms of the generators R and S (seeformula (2.5)) as follows A = R c S R c S · · · R c n S, where c i are some positive integers. The following statement is actually equivalent to Theorem 3.7. Corollary 6.8.
One has R c S R c S · · · R c n S = Id in PSL(2 , Z ) , if and only if ( c , . . . , c n ) is the quiddity of a d -dissection of an n -gon. Note that all of the above relations follow from the following two: S = Id , ( RS ) = Id , since PSL(2 , Z ) is known to be isomorphic to the free product of two cyclic groups with the generators S and RS , namely PSL(2 , Z ) ≃ ( Z / Z ) ∗ ( Z / Z ). Example 6.9.
We go back to Example 6.2.(a) Consider first the matrix A ′ = (cid:18) − − (cid:19) . It satisfies the condition from Part (ii) of Theorem 6.3,and, indeed, we see that 133 = [4 ,
3] = J , , K . One then checks that A ′ = M (5 , , A = (cid:18)
10 33 1 (cid:19) is as in Corollary 6.5, Part (i). Since = [3 ,
3] = J , , K , one easilychecks that A = M (4 , , , , , A = (cid:18) (cid:19) is as in Corollary 6.5, Part (ii). Since = [1 , , ,
1] = J , , K , onechecks that A = M (2 , , , , A is thus parametrized by (6 , A : x + = − √ J , , K . .6.3. Proof of Theorem 6.3 and Proposition 6.4.
Theorem 6.3 Part (i). Consider the following two“local surgery” operations.(1) Whenever the n -tuple ( c , . . . , c k ) contains a fragment c i , , c i +2 with c i , c i +2 >
1, the following“Conway-Coxeter operation” removes 1 and decreases the two neighboring entries by 1:(6.3) ( c , . . . , c i , , c i +2 , . . . , c k ) ( c , . . . , c i − , c i +2 − , . . . , c k ) . (2) Whenever the n -tuple ( c , . . . , c k ) contains a fragment c i , , , c i +3 (with arbitrary c i , c i +3 ), thefollowing operation reduces k by 3 producing the ( k − c , . . . , c i , , , c i +3 , . . . , c k ) ( c , . . . , c i + c i +3 − , . . . , c k ) . Note that these operations have already been used in Sections 3.1 and 3.2. It has already been checked,that these operations preserve the element M ( c , . . . , c k ) of PSL(2 , Z ).Given an arbitrary presentation A = M ( c , . . . , c k ) with positive integers c i , applying the opera-tions (6.3) and (6.4) when this is possible (and in arbitrary order), the k -tuple ( c , . . . , c k ) can be reducedto one of the case c i ≥
2, except perhaps for c , or c , c and c k , or c k − , c k . By multiplying from the right by M (1) or M (1 , c , . . . , c k ) in such a way that it contains at most one entry 1 at the end, i.e., that c k − ≥
2. Thisimplies that the continued fraction J c , . . . , c k K is well defined.Assume there are two different presentations of the same element, A = M ( c , . . . , c k ) = M ( c ′ , . . . , c ′ k ) , with positive integers c i and minimal k , so that the continued fraction J c ′ , . . . , c ′ k K = J c , . . . , c k K is alsowell defined. Then, without loss of generality, we can assume that c > c ′ . Since, for k ≥
3, one has J c , . . . , c k K > c − J c ′ , . . . , c ′ k K < c ′ , this implies that J c , . . . , c k K > J c ′ , . . . , c ′ k K , which contradictsthe assumption.Theorem 6.3 Part (ii). The positivity of the coefficients of A and the condition det A = 1 imply that ac < bd . We give a geometric argument using results of Section 1. The rationals ac < bd are linked by anedge in the Farey graph and one has the following two possible local pictures in T ac ∪ T bd . b b b b b bb a ′′ c ′′ ac bd . . . v ℓ +1 v ℓ +2 v k − v k − bb bbb bdac v k +2 b b . . . b b v j − a ′ c ′ b ′ d ′ b ′′ d ′′ The cases split as follows: in the left case a < b whereas in the right case a > b . In the rightcase bd is the previous convergent in the expansion of ac as negative continued fraction. In other words,if ac = J c , . . . , c k K then bd = J c , . . . , c k − K ; which gives A = M ( c , . . . , c k ) according to Proposition 2.1.Theorem 6.3 is proved.To prove Proposition 6.4, first note that the operations (6.3) and (6.4) allow one reduce any presentation A = M ( c , . . . , c k ) to the form with c i ≥
2, except perhaps for the ends of the sequence. Hence the “onlyif” part. The proof of the “if” part is similar to that of Theorem 6.3, Part (i).6.4.
Minimal presentation and Farey n -gon. Given a matrix A ∈ PSL(2 , Z ), we explain how torecover the coefficients ( c , . . . , c k ) in a combinatorial manner. More precisely, the coefficients can beinterpreted as the quiddity of some triangulated polygon.We focus on the case of matrices of the form A = a − bc − d ! with a, b, c, d >
0. Such a matrix A defines two (positive) rationals ac < bd that are linked by an edge inthe Farey tessellation. Similarly to what was done in Section 1.6, one draws two vertical lines from ac and bd in the Farey tessellation and collects all the triangles crossed by these lines. One thus obtains atriangulated Farey n -gon that we denote T A .Note that T A is the union T ac ∪ T bd of the triangulations defined in Section 1.2.We label the vertices of T A in decreasing order so that v = 10 , . . . , v k = bd , v k +1 = ac , . . . , v n = 01 , AREY BOAT 33 bbb b b b b b b b b b b
01 11 54 43 75 32 85 53 74 21 73 52 83 31 72 41 51 10 b b b b b
Figure 2.
The triangulation T A .and we denote by ( c , . . . , c k ) the quiddity sequence attached to the first k vertices. One has A = M ( c , . . . , c k ) . Example 6.10.
For the matrix A = (cid:18) − − (cid:19) , we obtain the triangulation of Figure 2.The corresponding Farey hexagon is (cid:0) , , , , , (cid:1) and the triangulation T A can be pictured asfollows: ☛☛ ✸✸✸✸✸✸✸✸ ✸✸ ✸✸ ☛☛
11 21
The quiddity sequence at the vertices , , is (3 , ,
1) so that we deduce A = (cid:18) − − (cid:19) = M (3 , , . Further examples: Cohn matrices.
Let us give more examples of interesting matrices.
Example 6.11. (a) Recall that an element A ∈ PSL(2 , Z ) is called parabolic if tr ( A ) = 2; a parabolicelement is conjugate to R a with a ∈ Z . The parabolic element R a , for a ≥ R a = M ( a + 1 , , , while the minimal presentations of L a and R − a are as follows: L a = (cid:18) a (cid:19) = M (1 , , . . . , | {z } a , , , R − a = (cid:18) − a (cid:19) = M (1 , , , . . . , | {z } a , . Note that the above equality hold in
P SL (2 , Z ), i.e., the matrix equalities are up to a sign. The elements L a and R − a belong to the same conjugacy class parametrized by (2 , , . . . , R a belongsto a different conjugacy class.(b) The minimal presentation of the continued fraction matrix M + ( a , . . . , a m ) is given by (2.7).(c) The famous Cohn matrices are the triples of matrices, (
A, AB, B ) in which the triples of Markovnumbers appear both, as right upper entry, and as of the traces. It is known (see [2]) that such matrices are enumerated by ( n, t ), where n ∈ Z and t is a rational 0 ≤ t ≤
1. The initial triple of Cohn matricesgiven by A ( n ) = n n − n − − n ! and B ( n ) := A ( n ) A ( n + 1) corresponds to the Markov triple (1 , ,
2) = (cid:0) tr ( A ) , tr ( AB ) , tr ( B ) (cid:1) .Other triples of Cohn matrices are given by the products of the matrices from the initial triple, encodedby a tree isomorphic to the Farey (or Stern-Brocot) tree of rationals in [0 , , , n ≥ A ( n ) = M (1 , , n − , , . . . , | {z } n , , ,B ( n ) = M (1 , , n − , , , . . . , | {z } n , , ,A ( n ) B ( n ) = M (1 , , n − , , , , . . . , | {z } n , , . Furthermore, A ( n ) B ( n ) = M (1 , , n − , , , , , . . . , | {z } n , , ,A ( n ) B ( n ) = M (1 , , n − , , , , , , . . . , | {z } n , , , etc.We see that all of these matrices with different n are conjugate to each other, the conjugacy classesof A and B being parametrized by (3) and (4 , The d -dissection of a matrix. We apply Theorem 6.3 in order to associate a 3 d -dissection toevery element A ∈ SL(2 , Z ). Our construction is as follows.Writing A and A − in the canonical minimal form A = M ( c , . . . , c k ) and A − = M ( c ′ , . . . , c ′ ℓ ) , oneobtains a ( k + ℓ )-tuple of positive integers ( c , . . . , c k , c ′ , . . . , c ′ ℓ ). Since M ( c , . . . , c k , c ′ , . . . , c ′ ℓ ) = M ( c , . . . , c k ) M ( c ′ , . . . , c ′ ℓ ) = Id , Theorem 3.7 implies that this is a quiddity of some 3 d -dissection. Example 6.12. (a) The matrix S = M (1 , , , ,
1) corresponds to the quiddity of the hexagonal dissec-tion of a decagon: 2 ❥❥❥❥ ❚❚❚❚ ✠✠ ✻✻ ✻✻ ✠✠ ❚❚❚❚ ❥❥❥❥ R one has R = M (2 , ,
1) (up to a sign) and R − = M (1 , , , ❖❖♦♦ ✎✎ ✴✴ ✴✴ ✎✎ AREY BOAT 35 (c) Consider the following elements A = (cid:18) (cid:19) , B = (cid:18) (cid:19) which are the simplest Cohn matrices (with n = 2). One has the following presentations: A = M (2 , , , , B = M (3 , , , , , A − = M (1 , , , , B − = M (1 , , , , . The corresponding quiddities are those of the dissected octagon and decagon:2 ☎☎ ❘❘❘❘❘❘❘ ✿✿ ✿✿ ☎☎ ❥❥❥❥ ❚❚❚❚ ✠✠✔✔✔✔✔✔ ✻✻ ☎☎☎☎☎☎☎☎☎✻✻ ✠✠ ❚❚❚❚ ✒✒✒✒✒✒✒✒✒✒✒✒ ❥❥❥❥ Acknowledgements . We are grateful to Charles Conley, Vladimir Fock, Sergei Fomin, Alexey Klimenko,and Sergei Tabachnikov for multiple stimulating and enlightening discussions. We are grateful to thereferee for a number of helpful remarks and suggestions. This paper was partially supported by the ANRproject SC3A, ANR-15-CE40-0004-01.
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Sophie Morier-Genoud, Sorbonne Universit´e, Universit´e Paris Diderot, CNRS, Institut de Math´ematiquesde Jussieu-Paris Rive Gauche, F-75005, Paris, FranceValentin Ovsienko, Centre national de la recherche scientifique, Laboratoire de Math´ematiques U.F.R.Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 Reims cedex 2, France
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