A FAMILY OF QUATERNIONIC MONODROMY GROUPS OF THE KONTSEVICH–ZORICH COCYCLE

A BSTRACT . For all d belonging to a density-1/8 subset of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group SO ∗ (2 d ) in its standard representation as the Zariski-closure of a factor of its monodromy. We prove that this conjecture holds for the ﬁrst elements of this subset, showing that the group SO ∗ (2 d ) is realizable for every 11 ≤ d ≤ 299 such that d = 3 mod 8, except possibly for d = 35 and d = 203.


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A translation surface (X, ω) is a compact Riemann surface equipped with a non-zero Abelian di erential.Away from its zeroes, ω induces an atlas on X all whose changes of coordinates are translations, called a translation atlas.Translation surfaces can be packed into a moduli space endowed with a natural SL(2, R)-action, given by post-composing with the coordinate charts of the translation atlas.The geometric and dynamical properties of this action have been extensively studied.We refer the reader to the surveys by Forni-Matheus [FM14], Wright [Wri15] and Zorich [Zor06] for excellent introductions to the subject.Integrating over ω provides coordinate charts for the moduli space of translation surfaces, called period coordinates.An a ne invariant manifold M is an immersed connected suborbifold of the moduli space of translation surfaces which is locally de ned by linear equations having real coe cients and zero constant terms in period coordinates.By the landmark work of Eskin-Mirzakhani [EM18] and Eskin-Mirzakani-Mohammadi [EMM15], a ne invariant manifolds coincide with orbit closures of the SL(2, R)-action.The Hodge bundle is the vector bundle over M whose bres are the homology groups H 1 (M; R), where M is the underlying topological surface of the elements of M. The Gauss-Manin connection provides a natural way to compare bres of the Hodge bundle.The Kontsevich-Zorich cocycle over M is the dynamical cocycle over the Hodge bundle induced by the SL(2, R)-action.This cocycle is at for the Gauss-Manin connection.The monodromy group of M is the group arising from the action of the (orbifold) fundamental group of M on the Hodge bundle.These groups can be also de ned by the action of the Kontsevich-Zorich cocycle on an SL(2, R)-invariant subbundle of the Hodge bundle.Moreover, the Kontsevich-Zorich cocycle is semisimple and its decomposition respects the Hodge structure [Fil16].Using these facts, Filip [Fil17] showed that the possible Zariski-closures of the monodromy groups arising from SL(2, R)-(strongly-)irreducible subbundles, at the level of real Lie algebra representations and up to compact factors, belong to the following list: (ii) su(p, q) in the standard representation; (iii) su(p, 1) in an exterior power representation; (iv) so * (2d) in the standard representation; or (v) so R (n, 2) in a spin representation.Nevertheless, it is not known whether every Lie algebra representation in this list is realizable as a monodromy group [Fil17,Question 1.5].Indeed, it is well-known that every group in the rst item is realizable.The groups in the second item were shown to be realizable by Avila-Matheus-Yoccoz [AMY17].Moreover, the group SO * (6) in its standard representation (which coincides with SU(3, 1) in its second exterior power representation) is also realizable by the work of Filip-Forni-Matheus [FFM18].
The main theorem of this article is the following: Theorem 1.1.For each d belonging to a density-1/8 subset of the natural numbers, there exists a square-tiled surface conjecturally realizing the group SO * (2d) as the monodromy group of an SL(2, R)-strongly-irreducible piece of its Kontsevich-Zorich cocycle.This conjecture depends on certain linear-algebraic conditions, which can be computationally shown to be true for small values of d.In this way, we show that SO * (2d) is realizable for every 11 ≤ d ≤ 299 in the congruence class d = 3 mod 8, except possibly for d = 35 and d = 203.Indeed, as was done by Filip-Forni-Matheus [FFM18], we will show that these groups seem to arise in quaternionic covers of simple square-tiled surfaces.This article is organized as follows.In Section 2, we cover the required background on monodromy groups of square-tiled surfaces.Section 3 shows the construction of the explicit family of square-tiled surfaces arising as quaternionic covers.Finally, we compute the desired monodromy groups in Section 4.

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2.1.Monodromy groups.Monodromy groups are, in general, a way to encode how a space relates to its universal cover.In the case of the Kontsevich-Zorich cocycle, they are de ned as follows: given an a ne invariant manifold (or orbit closure) M, we de ne its monodromy group as the image of the natural map π 1 (M) → Sp(H 1 (M; R)), where M is an underlying topological surface and π 1 (M) is the orbifold fundamental group.This means that they encode the homological action of the mapping classes identifying di erent points of the Teichmüller space to the same point of the moduli space.By the Hodge bundle we mean the vector bundle over M having H 1 (M; R) as the bre over every point.The Kontsevich-Zorich cocycle is the dynamical cocycle de ned over the Hodge bundle by the SL(2, R)-action.An SL(2, R)-invariant subbundle E is a subbundle for which g • E X = E g •X for every X ∈ M and g ∈ SL(2, R).A at subbundle E is a subbundle which is at for the Gauss-Manin connection.Observe that a at subbundle is necessarily SL(2, R)-invariant, since if the curvature vanishes then the parallel transport is done along SL(2, R)-orbits in the "obvious" way.The converse is not true in general: the atness condition requires no curvature in every possible direction, including those which are not reachable by the SL(2, R)-action.The Hodge bundle can be decomposed into irreducible pieces and monodromy groups can be de ned for such pieces.One then has the following [Fil17, Theorem 1.1]: Theorem 2.1.Let E be a strongly irreducible at subbundle of the Hodge bundle over some a ne invariant manifold M.Then, the presence of zero Lyapunov exponents implies that the Zariski-closure of the monodromy group has at most one non-compact factor, which, up to niteindex, is equal at the level of Lie group representations to • SU(p, q) in the standard representation • SU(p, 1) in any exterior power representation or • SO * (2d) in the standard representation for some odd d.Observe that this is a "re ned" version of the constraints in the previous section, under stronger hypotheses.
2.2.Square-tiled surfaces.A square-tiled surface is a particular kind of translation surface de ned as a nite cover of the unit square torus branched over a single point.That is, we say that a translation surface (X, ω) is square-tiled if there exists a covering map π : X → R 2 /Z 2 , which is unrami ed away from 0 ∈ R 2 /Z 2 , and ω = π * (dz), where dz is the Abelian di erential on R 2 /Z 2 induced by the natural identi cation R 2 C. We will often write X to refer to (X, ω) for simplicity.Combinatorially, a square-tiled surface can be de ned as a pair of horizontal and vertical permutations h, v ∈ Sym(Sq(X)), where Sq(X) is some nite set that we interpret as the squares of X.These two permutations can be obtained from our original de nition as the deck transformations induced respectively by the curves t → (t, 0) and t → (0, t), with t ∈ [0, 1], and the set of squares can be de ned to be Sq(X) = π −1 ((0, 1) 2 ).Conversely, we can glue squares horizontally using h and vertically using v and de ne ω to be the pullback of dz in each square to obtain a square-tiled surface as in the original de nition.
2.2.1.SL(2, R)-action and monodromy groups.Every square-tiled surface X is a Veech surface, that is, its SL(2, R)-orbit is closed.In particular, this implies that any SL(2, R)invariant subbundle of the Hodge bundle over the orbit SL(2, R) • X is actually at.Therefore, Theorem 2.1 can be applied for any SL(2, R)-(strongly-)irreducible subbundle.We say that square-tiled surface X is reduced if the covering map π cannot be factored through another non-trivial covering of the torus.In this case, the elements g ∈ SL(2, R) such that g • X is a square-tiled surface are exactly SL(2, Z).It is often the case that we study the action of SL(2, Z) on X instead of the entire SL(2, R)-action, since square-tiled surfaces can be represented in purely combinatorial terms.The Veech group of X, usually denoted SL(X), is the subgroup of SL(2, Z) stabilizing X.It is always an arithmetic subgroup of SL(2, Z) and its index coincides with the cardinality of SL(2, Z) • X.Every square-tiled surface that we will consider is reduced.A square-tiled surface may also have non-trivial automorphisms.In this case, the SL(2, Z)-action does not immediately induce a homological action on the Hodge bundle.Indeed, automorphisms are precisely the reason why orbit closures are, in general, orbifolds and not manifolds.More precisely, we de ne an a ne homeomorphism as an orientation preserving homeomorphism of X whose local expressions (with respect to the translation atlas) are a ne maps of R 2 .We denote the group of a ne homeomorphisms by A (X).We may extract the linear part of an a ne homeomorphism to get a surjective homomorphism A (X) → SL(X).The kernel of this homomorphism is the group Aut(X) of automorphisms of X.This can be encoded in the form of a short exact sequence: In other words, if M is the underlying topological surface of X, then Aut(X) is precisely the subgroup of Mod(M) stabilizing a lift of X to the Teichmüller space of translation surfaces.In this sense, it measures to which extent the Mod(M)-action fails to be free at X. Automorphisms can also be de ned combinatorically: they are the elements of Sym(Sq(X)) that commute with both h and v.It is well-known that if X has only one singularity, then it has no non-trivial automorphisms.
The homology group H 1 (M; R) admits a splitting H st 1 (M) ⊕ H (0) 1 (M) into symplectic and mutually symplectically orthogonal subspaces.The subspace H st 1 (M) is twodimensional and is usually called the tautological plane.It is spanned the following two cycles: the sum of all bottom horizontal sides of the squares of X oriented rightwards, and the sum of all left vertical sides of the squares of X oriented upwards.The subspace 1 (M) consists of the zero-holonomy cycles, that is, the cycles c such that ∫ c ω = 0. Let ρ : A (X) → Sp(H 1 (M; R)) be the representation induced by the homological action of A (X).By restricting this representation to an invariant subspace, we obtain a monodromy representation of a subbundle of the Hodge bundle.We de ne the monodromy group of this subbundle to be the image of this representation.The group ρ(A (X)) preserves the splitting Moreover, the space H st 1 (M) is also irreducible and its monodromy group is a nite-index subgroup of SL(2, Z) = Sp(2, Z) which can be identi ed with SL(X).The subspace H (0)  1 (M) is in general reducible.Therefore, understanding monodromy groups means understanding the irreducible pieces of H (0)  1 (M) and the way ρ(A (X)) acts on them.2.2.2.Constraints for monodromy groups.Let G = Aut(X).The vector space H 1 (M; R) has a structure of a G-module induced by the representation G → Sp(H 1 (M; R)).Since G is a nite group, it possesses nitely many irreducible representations over R which we denote Irr R (G).The G-module H 1 (M; R) can be decomposed as a direct sum of irreducible representations.That is: where each V α is an irreducible subspace of H 1 (M; R) on which G acts as the representation α.We can collect the same G-irreducible representations into the so-called isotypical components.That is, let W α = V ⊕n α α and then: The group ρ(A (X)) does not, a priori, respect this decomposition because a general a ne homeomorphism may not commute with every automorphism.However, since G is a nite group, there exists a nite-index subgroup of A * (X) ≤ A (X) whose every element commutes with every element of G. Replacing A (X) by some niteindex subgroup preserves the Zariski-closure of the resulting monodromy group.Given an irreducible representation α of G, we may de ne an associative division algebra D α : the centralizer of α(G) inside End R (V α ).Up to isomorphism, there are three associative real division algebras: • D α R, and α is said to be real; • D α C, and α is said to be complex; or • D α H, and α is said to be quaternionic.The following theorem [MYZ14, Section 3.7; MYZ16] relates these cases to constraints for monodromy groups: Theorem 2.2.The Zariski-closure of the group ρ • SU(p α , q α ) if α is complex or • SO * (2d α ) if α is quaternionic.We will exploit these constraints to nd the desired groups.

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In this section, we will construct the quaternionic covers that realize the desired groups as the Zariski-closure of the monodromy of a speci c at irreducible subbundle of the Hodge bundle.Let d ≥ 3 be an odd integer.We consider a "staircase" X (d) with d squares: the square-tiled surface induced by the permutations (2, 1)(4, 3) . . .
Its automorphism group is trivial, since it belongs to a minimal stratum.We construct a cover X (d) of X (d) as follows: for each element g of the quaternion group Q = {1, −1, i, −i, j, − j, k, −k}, we take a copy X (d) g of X (d) .We glue the r-th right vertical side of X (d)  g to the r-th left vertical side of X (d)  gi .Similarly, we glue the r-th top horizontal side of X (d)  g to the r-th bottom horizontal side of X (d) g j .See Figure 1.This construction coincides, up to relabelling, with that of Filip-Forni-Matheus for d = 3 [FFM18, Section 5.1].For each g ∈ Q, we can de ne an automorphism ϕ g of X (d) by mapping X (d)  h to X (d)   gh in the natural way, that is, preserving the covering map X (d) → X (d) for each h ∈ Q.Indeed, the gluings are de ned by multiplication on the right, which commutes with multiplication on the left.These are the only automorphisms of X (d) : an automorphism ψ of X (d) induces an automorphism of X (d) by "forgetting the labels".Since the only 2. An illustration of X (3) showing its four singularities.
automorphism of X (d) is the identity, X (d)  1 is mapped to some X (d) g for g ∈ Q in a way that preserves the covering map X (d) → X (d) .Thus, ψ = ϕ g and Aut( X (d) ) Q.We will denote Aut( X (d) ) by G. From now on, we will restrict to the case d = 3 mod 8.The surface X (d) has four singularities, each of order 2d − 1.Therefore, X (d) belongs to the (connected) stratum H 4d−1 ((2d − 1) 4 ).Since the automorphism ϕ −1 ∈ G is an involution, it induces a splitting , where where ϕ −1 acts as ±Id.These subspaces are symplectic and symplectically orthogonal.The subspace H + 1 ( X (d) ) contains H st 1 ( X (d) ) and is naturally isomorphic to H 1 (X (d)  ± ; R), where X (d) ± = X (d) /ϕ −1 .This latter surface is an intermediate cover of X (d) over the group Q/{1, −1} Z/2Z × Z/2Z.Since every singularity of X (d) is xed by ϕ −1 , X (d)  ± belongs to the stratum d) ; R) and we obtain that the dimension of H − 1 ( X (d) ) is 4d.The irreducible representations (over C) of the group Q can be summarized in the following character table: As detailed in Section 2.2.2, H 1 ( X (d) ; R) can be split into isotypical components associated with such representations.From the character table, we obtain that H − 1 ( X (d) ) corresponds to 2d copies of a G-irreducible representation whose character is the quaternionic character χ 2 , that is, H − 1 ( X (d) ) = W χ 2 .Indeed, ϕ −1 acts as the identity for any other representation in the table.We obtain the following: Lemma 3.1.The Zariski-closure of the monodromy group of the at subbundle induced by H − 1 ( X (d) ) is a subgroup of SO * (2d).Moreover, Kontsevich-Zorich cocycle over this subbundle has at least four zero Lypaunov exponents.
Proof.The rst statement is a direct consequence of Theorem 2.2.The second statement is a consequence of the rst since d is odd [Fil17, Corollary 5.5, Section 5.3.4].
We will prove that, for certain d with d mod 8 = 3, such Zariski-closure is actually SO * (2d).

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4.1.Dimensional constraints.In the presence of zero Lyapunov exponents, Theorem 2.1 states that the only possible Lie algebra representations of the Zariski-closure of the monodromy group of a at are so * (2d) in the standard representation, su(p, q) in the standard representation and su(p, 1) in some exterior power representation.The exterior power representations of su(p, 1) are irreducible and faithful: by complexifying, one obtains sl(p+1, C) whose exterior power representations are known to satisfy these properties.Let p + 1 r for every p and 1 < r < p .
The following lemma shows that, assuming irreducibility, so * (2d) is the only possible Lie algebra for every d ∈ D. By a slight abuse of notation, we will denote the standard representation of so * (2d) by so * (2d), the standard representation of su(p, q) by su(p, q) and the r-th exterior power of the standard representation of su(p, 1) by Λ r (su(p, 1)).Lemma 4.1.Assume that so * (2d), su(p, q) and Λ r (su(p, 1)) act irreducibly on a vector space of the same dimension, that is, 2d = p + q = p+1 r .Then, for every d ∈ D we have that dim R so * (2d) < dim R su(p, q) and that dim R so * (2d) < dim R Λ r (su(p, 1)).
Remark 4.2.To obtain the strict inequality in the previous proof, it is necessary to assume that d ∈ D. Indeed, if d D then 2d = p+1 r with 1 < r < p.This implies that dim R Λ r (su(p, 1)) ≤ dim R so * (2d) since 2d = p+1 r ≥ p+1 2 , so p(p + 1) ≤ 4d and it is easy to check that this results in p(p + 2) ≤ d(2d − 1) if d ≥ 3.

Proof. Let
We 4.2.Dehn multi twists.We will use Dehn multi twists along speci c rational directions to prove irreducibility.Assume that there exist rational directions (p r , q r ) for 0 ≤ r < d such that: (1) the cylinder decomposition along (p r , q r ) consists of eight cylinders with waist curves c r g , for g ∈ Q, of the same length.Thus, the Dehn multi twist along (p r , q r ) can be written as T r v = v + n r g ∈G v, c r g c r g ; and (2) the action of G on the labels is "well-behaved", that is, (ϕ h ) * c r g = c r hg for every 0 ≤ r < d, and g, h ∈ Q.
Proof.Let V {0} be a subspace of H − 1 ( X (d) ) on which A * * ( X (d) ) acts irreducibly.By (3), it is enough to prove that C r ⊆ V for each 0 ≤ r < d.Since the index of A * * ( X (d) ) is nite, some power of T r belongs to A * * ( X (d) ) for every 0 ≤ r < d.Without loss of generality, we can assume T r ∈ A * * ( X (d) ), since the number n r 0 in the formula for T r is irrelevant for the proof.We will rst show that By (2), we have that We obtain that (ϕ g ) * v g ∈Q + = C r ⊆ V , which completes the proof.
We can now show that this conditions are enough for the monodromy group to be SO * (2d): Proposition 4.5.Assume that d = 3 mod 8, that d ∈ D and that (1)-(5) hold.Then, the Zarisk-closure of the group ρ(A * (X))| H − 1 ( X (d) ) is SO * (2d).Proof.By Lemma 3.1, exactly four Lyapunov exponents of the Kontsevich-Zorich cocycle are zero, so the hypotheses of Theorem 2.1 are satis ed.To conclude by Lemma 4.1, it is enough for A * (X) to act strongly irreducibly on H − 1 ( X (d) ), which follows from the previous lemma.
The next section is then devoted to nding the desired Dehn multi twists.4.3.Suitable rational directions.In this section, we will nd the desired rational directions (p r , q r ) and prove (1)-(5) for the speci c values of d mentioned in the statement of the main theorem to conclude the proof.Assume that d = 3 mod 8 for the rest of the section.The matrices generate SL(2, Z) and, thus, can be used to understand the SL(2, Z)-orbit of a squaretiled surface.The orbit of the "staircase" X (d) consists of three elements, which we call Z (d) , X (d) and Y (d) .See Figure 3.

T S
T 3. The SL(2, Z)-orbit of X (d) using T and S as generators.It consists of three distinct square-tiled surfaces, which we call Z (d) , X (d)  and Y (d) from left to right.The labels in the Y (d) and Z (d) show the identi cation of the sides.Unlabelled horizontal sides are identi ed with the only horizontal having the same horizontal coordinates, and similarly for unlabelled vertical sides.
Since X (d) is a cover of X (d) , the graph induced by the action on T and S on X (d) is a cover of the graph in Figure 3.In other words, if g ∈ SL(2, Z) then g • X (d) is a degreeeight cover of g • X (d) .Moreover, since the graph in Figure 3 has only three vertices, writing g in terms of T and S and following the arrows of the graph us to compute g • X (d) , which is useful to understand g • X (d) .We will use the following rational directions: (p r , q r ) = (−(4r + 1), 4r + 3) for 0 ≤ r < d.
Observe that the matrix 2r + 1 2r 4r + 3 4r + 1 maps the direction (p r , q r ) to (−1, 0).Moreover, this matrix can be written as S 2 T 2r S. By Figure 3, S 2 T 2r S • X (d) = Y (d) , so this surface has only one horizontal cylinder.The matrix S maps (p r , q r ) to (−(4r + 1), 2).The surface S • X (d) , which we call Y (d) , is a degree-eight cover of S • X (d) = Y (d) , which we will now describe explicitly.
For each g ∈ Q, consider a copy Y (d) g of Y (d) .Each of these copies consists of d squares.We label the r-th bottom side of each square of Y (d)  g with η r g and the left side of the leftmost square with ζ g .Let m = (d + 1)/2, which satis es m = 2 mod 4 since d = 3 mod 8.There are m − 1 squares to the left and to the right of m in Y (d)  g .The labels of the top sides of the squares to the right of m are: gk .The labels of the top sides of the squares to the left of m are: . ., η m+3 gi , η m+2 −g j , η m+1 −gi , η m g j .In the two previous lists, the group elements in Q follow a 4-periodic pattern.Finally, we label the rightmost square of Y (d)  g with ζ −g .See Figure 4 for an illustration.By a slight abuse of notation, from now on we will use the names η r g and ζ g to refer to the elements of H 1 ( Y (d) , Σ; R) induced by the horizontal or vertical curves joining the An illustration of Y (d)  g and of the cut-and-paste operations used to obtain this description.
two vertices of the side labelled η r g or ζ g , oriented either rightwards or upwards.We have that Aut( Y (d) ) Q, which can be proved in the exact same way as for X (d) .That is, we de ne an automorphism ϕ g by mapping Y (d)  h to Y (d)  gh and these are the only automorphisms of Y (d)  d) ).The space H − ( Y (d) ) is 4d-dimensional and it is exactly the image of H − ( X (d) ) by S. Let ηr g = η r g − η r −g for g ∈ Q + = {1, i, j, k} and 1 ≤ r ≤ d.We have that each ηr g is an absolute cycle since ϕ −1 xes every singularity.Therefore, ηr g ∈ H − 1 ( Y (d) ) and we obtain that { ηr g } g ∈Q + ,1≤r ≤d is a basis of H − 1 ( Y (d) ).Observe that Y (d) has four horizontal cylinders.The matrix T 2r maps the direction S(p r , q r ) = (−(4r + 1), 2) to (−1, 2).Therefore, understanding the direction (p r , q r ) on X (d) is equivalent to understanding the direction (−1, 2) on T 2r • Y (d) .We will start the analysis for r = 0.For g ∈ Q, consider the trajectory induced by the direction (−1, 2) on Y (d)  g as in Figure 5.The resulting cylinder decomposition consists on eight cylinders.Indeed, observe that each cycle η r g is intersected twice by such trajectories.Therefore, the total number of intersections of all the η r g by all trajectories is 16d.To obtain that there are exactly eight cylinders in this decomposition, it is therefore enough to show that each trajectory intersects exactly 2d cycles η r g .The trajectory in Figure 5 intersects the following cycles: g .The gluings are cyclically shifted and the signs of elements of Q on the labels η 1 • are changed.
where the sequence g 1 , . . ., g 2d+1 is obtained by (right-)multiplying g successively by The boxed comes from the intersection with the vertical side labelled as ζ g d .This sequence indeed describes a closed trajectory as g 2d+1 = g.Indeed, the product can be computed from "inside out", using that −1 is in the centre of Q.We obtain that Moreover, the number of times 1 2 and (−1) 2 occur in this product is (m − 2)/2, which is an even number as m mod 4 = 2, and the total number of terms is d + 1, which is also an even number.Thus, g 2d+1 = g and we conclude that the cylinder decomposition induced by (−1, 2) has exactly eight cylinders.Moreover, we obtain that the action of Aut( Y (d) ) on these waist curves is "well-behaved" in the sense of (2): naming the trajectory starting on Y (d) g as c 0 g , we get that (ϕ h ) * c 0 g = c 0 hg .Now, if r = 1 then Y (d)  g is sheared horizontally in such a way that the labels η m+1 g and η m+1 −gi end up on the same square.We will consider this square to be the "middle" square and reglue the surface accordingly.The surface T 2 • Y (d) is the union of sheared and reglued versions of Y (d)  g , for g ∈ Q, that we call T 2 •Y (d) g .See Figure 6 for an illustration.
In general, T 2r • Y (d) , for 0 ≤ r < d, is the surface obtained from Y (d) by cyclically shifting the labels on the top sides r times to the right, the ones on the bottom sides r times to the left, and changing the signs of the elements of Q of every label of the form η s • for 1 ≤ s ≤ r.We conclude that the cylinder decomposition of T 2r • Y (d) induced by the direction (−1, 2) consists of exactly eight cylinders in the same way as for the case r = 0 and denote their waist curves by c r g .The action of G is then well-behaved in the sense of (2).By construction, (1) also holds.Let ĉr g = c r g − c r −g for g ∈ Q + .It remains to prove (3), (4) and (5) to conclude the proof.We conjecture that these two conditions hold for every d belonging to the congruence class d = 3 mod 8. Nevertheless, the previous discussion allows us to compute the intersection numbers explicitly using a computer.Indeed, to obtain (3) we can compute the numbers ĉr g , η s h for each 0 ≤ r, s ≤ d.Then, we can compute the determinant of the resulting matrix to show that it is not singular.This matrix also allows us to compute ĉr g , ĉs h by expressing each ĉr g in terms of the basis { ηr g } g ∈Q + ,1≤r ≤d to show (4) and (5).

Lemma 4. 3 .
The set D has full density in N.