Symmetry, Integrability and Geometry: Methods and Applications Period Matrices of Real Riemann Surfaces and Fundamental Domains

For some positive integers $g$ and $n$ we consider a subgroup $\mathbb{G}_{g,n}$ of the $2g$-dimensional modular group keeping invariant a certain locus $\mathcal{W}_{g,n}$ in the Siegel upper half plane of degree $g$. We address the problem of describing a fundamental domain for the modular action of the subgroup on $\mathcal{W}_{g,n}$. Our motivation comes from geometry: $g$ and $n$ represent the genus and the number of ovals of a generic real Riemann surface of separated type; the locus $\mathcal{W}_{g,n}$ contains the corresponding period matrix computed with respect to some specific basis in the homology. In this paper we formulate a general procedure to solve the problem when $g$ is even and $n$ equals one. For $g$ equal to two or four the explicit calculations are worked out in full detail.


Introduction
The Siegel upper half plane of degree g, usually denoted H g , is defined as the space of g × g complex, symmetric matrices whose imaginary part is positive definite. The modular group of dimension 2g is the group of 2g × 2g symplectic matrices whose entries are integer numbers. We will denote it by the symbol Sp 2g, Z . There exists a well-known action of the modular group on the Siegel upper half plane, the so called modular action: To every matrix for some non-negative integer p. Let us define the space W g,n = w ∈ H g such that VwV = −w where the matrix V is given by Let us also introduce the group G g,n = G ∈ Sp 2g, Z such that GT = TG where T denotes the 2g × 2g matrix The modular action can be restricted to an action of G g,n on the space W g,n : for every matrix G = P Q R S ∈ G g,n and every point w ∈ W g,n one has that 1 M (G, w) ∈ W g,n .
We wish to address the problem of a fundamental domain for this action of the group G g,n ⊂ Sp 2g, Z on the space W g,n . Its definition is recalled here below: Definition 1.1. We say that two points w 1 and w 2 of W g,n are equivalent if there exists G ∈ G g,n such that w 1 = M (G, w 2 ) .
A fundamental domain for M is a closed subset D of W g,n whose interior part is connected, which satisfies the following three properties: i For every w ∈ W g,n there exists an equivalent pointw belonging to D.
ii If two distinct points, w andw, belonging to D, are equivalent, then they both belong to the boundary of D.
iii Every set of equivalent points contained in D has a finite number of elements.
The case in which n equals g + 1 is easily reconducted to the Reduction Theory of positive definite quadratic forms and was already studied by several mathematicians from the last century. (The first achievements in a general treatment of this topic are due to Minkowski [1]; see [10] for an account on more recent results.) In this article we solve the problem when g = 2g 0 is an even number, and n equals one. In this case, the space W 2g 0 ,1 has the following simple characterization: For the group G 2g 0 ,1 , instead, one has It is understood that the matrices A, B, C, D, E, F, G and H belong to Mat g 0 , Z .
Our main motivation in considering this problem comes from the theory of real Riemann surfaces (see [3] and [5] for a complete account of this subject). A compact Riemann surface Γ of genus g is said to be real or symmetric when it is endowed with an anti-holomorphic involution r. One can consider the locus of points which are invariant with respect to r. Its connected components are at most g + 1 and they are usually called the ovals of the surface. When Γ with this locus removed has two connected components, it is said to be separated. Let this be the case, and assume 2 that the number of ovals is n. Then it is possible to fix a basis in the homology of the form a ′ 1 , a ′ 2 , . . . , a ′ p , a ′′ 1 , a ′′ 2 , . . . , a ′′ p , a 1 , a 2 , . . . , a n−1 satisfying the following conditions: for every k = 1, 2, . . . , p and every j = 1, 2, . . . , n − 1, the intersection number of any two other elements of the basis being zero. 2 In the case of a separated surface one has that 1 ≤ n ≤ g + 1 and n ≡ g + 1 mod 2.
r ⋆ a j = −a j , j = 1, 2, . . . , n − 1; where r ⋆ denotes the morphism of the homology into itself induced by r.
Given any Riemann surface (not necessrily a real one) with a fixed basis in the homology B, one can compute the corresponding period matrix 3 . This is a point in the Siegel upper half plane (see [2] and [5] for more details). Any other basis in the homologyB on the same surface can be expressed in terms of the cycles of B by means of a modular matrix. If this is blockwise written as then the corresponding period matrices t andt are related by the modular transformationt Now, in the case of a real Riemann surface with a basis B of the form (3)(4)(5), the period matrix belongs to the space W g,n . Moreover, the changes of basis in the homology which preserve the form (3)(4)(5) are exactly the ones given by matrices of G g,n .
Real Riemann surfaces were extensively applied in the theory of integrable systems (see [11]). In [6], the solution of our problem in one particular case was successfully applied to the study of the parameter space of algebro-geometric solutions of the KP-2 equation.
As we have already said, in this paper we formulate a general procedure to determine a fundamental domain for the modular action of the group G 2g 0 ,1 on the space W 2g 0 ,1 . We briefly sketch our methods and results here below: Our first step consists in a reformulation of the problem (Section 2). Let us denote with GL g, Z the group of all g-dimensional, unimodular matrices with integer entries. Let us also introduce the space Sym >0 g, R of all real symmetric and positive definite matrices of dimension g. The congruent action 3 In this procedure, the normalized holomorphic differentials are involved. Throughout this paper we will use the convention for k = 1, 2, . . . , p and j = 1, 2, . . . , n − 1.
is defined as follows The modular and the congruent actions are closely related by a result of Siegel's (Theorem 2.1): There exists an injective and smooth map such that the following diagram is commutative for every G belonging to Sp 2g, Z H g Let us define the space The congruent action can be restricted to an action of the group G 2g 0 ,1 on S 2g 0 ,1 . Now, let D ′ be a fundamental domain for this last one, and put Using Siegel's result one can show that the set D so defined is a fundamental domain for the modular action restricted to G 2g 0 ,1 and W 2g 0 ,1 . The original issue can then be reformulated as the quest of such a D ′ .
The main technical tool to tackle this new problem is the map P, introduced in Section 3. Due to its several remarkable properties this might turn out to be of general interest by itself, beyond its role in this specific context. To the best of our knowledge, this map was never considered in the literature before.
The domain of P is the set of all 4g 0 -dimensional, real and symplectic matrices of the form Both S 2g 0 ,1 and G 2g 0 ,1 are contained in it. Its explicit definition is very simple: After this, P turns out to be a bijection onto GL 2g 0 , R , the space of all real and invertible matrices of dimension 2g 0 . It maps S 2g 0 ,1 onto Sym >0 2g 0 , R .
Moreover P respects both the matrix product and transposition. As a consequence the following diagram is commutative for every G belonging to G 2g 0 ,1 : This construction allows a further reformulation of the problem, in the same spirit as above: Let us consider the group K 2g 0 ,1 defined as This is a finite-index subgroup of GL 2g 0 , Z . Obviously, the congruent action can be restricted to an action of K 2g 0 ,1 on Sym >0 2g 0 , R ; let D ′′ be a fundamental domain for it. Due to the properties of P the set D ′ defined as turns out to be a fundamental domain for the congruent action restricted to G 2g 0 ,1 and S 2g 0 ,1 . The original problem is then equivalent to the quest of such a D ′′ .
This second reformulation appears in Section 4. It is given the name of Reduction of the Problem because it halves, so to say, the dimension of the matrices involved in the issue.
The advantage of these subsequent reformulations is that we finally get to an explicitly solvable problem. Indeed, the Minkowski Reduction Theory provides us with a fundamental domain for the congruent action of the whole group GL 2g 0 , Z on Sym >0 2g 0 , R . On the other side, the index of K 2g 0 ,1 in GL 2g 0 , Z is finite. In view of these two facts, a fundamental domain D ′′ for the congruent action of K 2g 0 ,1 on Sym >0 2g 0 , R can be computed by means of standard "gluing" techniques.
The case in which 2g 0 equals 2 is particularly interesting (Section 5). It exhibits the remarkable peculiarity that the group K 2,1 coincides with the whole GL (2, Z). As a consequence, the two reformulations above reconduct the original problem to a classical one already solved: the reduction of positive definite binary quadratic forms. No usage of "gluing" techniques is required in this case. The formulas expressing the result are very simple: Let us denote by the generic element of W 2,1 ; a fundamental domain D for the modular action M of the group G 2,1 ⊂ Sp (4, Z) on this last one is given by the following system of inequalities: D : Moreover, a deeper understanding of the mathematical structure is possible in this case: Let G belong to G 2,1 . One has that either Introduce, on W 2,1 , the new system of coordinates

Now consider a modular transformatioñ
In the new coordinates it acts as follows: In other words, we find out that I is an invariant quantity: it is not affected by the modular action of G 2,1 , which concentrates only on the second coordinate τ. This last one just undergoes a Moebius transformation, in some cases composed with a complex conjugation. The theory developed in Sections 2 and 3 allowed us to individuate the higher dimensional analogue of the invariant quantity I and to prove that also in this case it is not affected by the modular action of G 2g 0 ,1 (Section 4). A full understanding of the geometrical meaning of this quantity has not yet been achieved though, and it is meant to be part of a work in progress. Finally, the explicit calculation of a fundamental domain has been worked out in the case when 2g 0 equals 4. The result is not so simple to write down; a rather detailed account of the formulas there involved can be found in Section 6.

Reformulation of the Problem
In this section we show how the original problem can be refomulated in terms of a restriction of the congruent action C, introduced in (6-7).
For every matrix w = λ + ıµ ∈ H g belonging to the Siegel upper half plane of degree g, let us define This map already appeared in Siegel's investigations on the modular group, from which we drew some inspiration. It relates the modular and the congruent actions: i Σ is a bijection onto the space of all real, symmetric and positive definite matrices of dimension 2g which are also symplectic.
ii For every G belonging to the modular group, the following diagram is commutative: For a proof see [4], pag. 148, theorem 1.
Now, since the original problem is concerned with the proper subset W 2g 0 ,1 ⊂ H 2g 0 , let us introduce the space In view of point ii of theorem 2.1, it is easy to realize that C can be restricted to an action of the group G 2g 0 ,1 on S 2g 0 ,1 . Moreover the following diagram is commutative, for every G belonging to G 2g 0 ,1 : This immediately gives the following

Corollary 2.2. Let D ′ be a fundamental domain for the congruent action of the group
is a fundamental domain for the modular action of the same group G 2g 0 ,1 on W 2g 0 ,1 .
In view of this, the original problem is equivalent to the quest of such a D ′ . To this purpose, a more explicit characterization of S 2g 0 ,1 will be useful: The set S 2g 0 ,1 consists of all real, symplectic, symmetric and positive definite matrices Σ of dimension 4g 0 , which have the following form for some α, β, γ, δ, ξ and η belonging to Mat g 0 , R .
Proof. In view of theorem 2.1, point i, it will be sufficient to prove that a matrix w of the Siegel upper half plane belongs to W 2g 0 ,1 if and only if its image Σ (w) has the form (17). This last condition is equivalent to the relation Using the explicit definition (14), one can show that (18) holds if and only if the following relations do: But this happens if and only if w belongs to W 2g 0 ,1 , so the proof is complete.
Next sections will be dedicated to determine a fundamental domain D ′ for the congruent action of G 2g 0 ,1 on S 2g 0 ,1 .

Reduction Toolkit
In this section we introduce some objects which will be, so to say, useful tools in our task.
Let us start with an auxiliary map. Consider the vector space It is understood that the matrices α, β, γ, δ, π, ρ, ξ and η belong to Mat g 0 , R . We define the map as follows: Let us also introduce the notation for the standard symplectic matrix. Its dimension will be clear from the context. Lemma 3.1. Let Σ and Σ ′ belong to V 2g 0 ,1 and put One has the following 4 : Proof. Points i, ii and iii can be verified by means of a straightforward calculation.
iv Since G is bijective (see point i,) one has that and Σ is symmetric if and only if both σ and τ are; let us suppose that this is the case. One can verify that the two identities The space V 2g 0 ,1 is closed with respect to matrix product. Indeed, a 4g 0 -dimensional, real matrix belongs to V 2g 0 ,1 if and only if it satisfies the relation where T is the idempotent matrix defined in (19). hold true for every column vector α, β belonging to R g 0 . This implies that if Σ is positive definite, then also σ and τ are. Viceversa, the quadratic form given by Σ can be expressed in terms of the quadratic forms corresponding to σ and τ in the following way: So, if both σ and τ are positive definite, then also Σ is.
v By definition, Σ is symplectic if and only if which, by carrying out the products in the left-hand side, reduces to the following system of matrix-equations: Summing and subtracting (22) and (23) one gets the equations which are equivalent one to the other. So (22) and (23) together are equivalent to the unique equation Analogously, (24) and (25) together are equivalent to Equations (26) and (27) are indeed equivalent to the system So Σ is symplectic if and only if it satisfies the following system of equations: (30) but after transposition of (29) and (30), this turns out to be equivalent to the condition Let us introduce the group 5 It is understood that the matrices α, β, γ, δ, π, ρ, ξ and η belong to Mat g 0 , R . Notice that both G 2g 0 ,1 and S 2g 0 ,1 are contained in it. Let us also define P :S 2g 0 ,1 −→ Mat 2g 0 , R as follows: Many properties of this map can be deduced from lemma 3.1: The map P is a homeomorphism fromS 2g 0 ,1 to GL 2g 0 , R , this last one consisting of all real invertible matrices of dimension 2g 0 . Moreover it respects both matrix product Proof. The inverse of P is the map Q : GL 2g 0 , R −→ S 2g 0 ,1 defined as follows: One can easily verify that this is a group using the argument quoted in footnote 4 Notice that the image of Q is contained inS 2g 0 ,1 because of point v of the lemma. Since both Q and P are continuous, this last one is a homeomorphism. Moreover P respects both matrix product and transposition due to point ii and iii of the lemma.
Proof. From point iv of lemma 3.1, one immediately has that Viceversa, let σ belong to Sym >0 2g 0 , R . The matrix From the proof of the previous corollary, we know that In view of point iv of the lemma, this last matrix is also positive definite and so it belongs to S 2g 0 ,1 . It follows that P S 2g 0 ,1 = Sym >0 2g 0 , R and the proof is complete.
One has the following Corollary 3.4. The map P restricts to an isomorphism of groups from G 2g 0 ,1 to K 2g 0 ,1 .
Proof. Let G belong to G 2g 0 ,1 and put From point v of the lemma, one has that Since both σ and τ have integer coefficients, this implies that Equation (32) can be rewritten as follows: On the other side, from definition (20) one has that for someM belonging to Mat 2g 0 , Z , in this case. Plugging (34) into (33) one obtains that for some M belonging to Mat 2g 0 , Z . Viceversa, let g belong to K 2g 0 ,1 . Let us put From the proof of corollary 3.2 we know that Recalling the definition of G one can easily write for some M belonging to Mat 2g 0 , Z , one also has that for someM belonging to Mat 2g 0 , Z . Since g has an inverse with integer entries, it follows immediately that for someÑ belonging to Mat 2g 0 , Z . Equations (37) together with (38) and (39) imply that P −1 g has integer coefficients. So it belongs to G 2g 0 ,1 .

Remark 3.5. The whole modular group
It is easy to show that the argument of the function P in the left-hand side belongs to G 2g 0 ,1 . Since Id 0 0 −Id ∈ K 2g 0 ,1 the following inclusion also holds: and the argument of P again belongs to G 2g 0 ,1 .

Reduction of the Problem
We can now reformulate further the original problem, using the instruments developed in the previous section. This process bears the name of reduction because it halves the dimension of the matrices involved in the issue.

Proposition 4.1. The following diagram
is commutative for every G belonging to G 2g 0 ,1 .
On the other side, simply by definition, As a consequence, one has the following The original problem is thus equivalent to the quest of such a D ′′ . The remaining sections will be dedicated to this issue. This will also emphasize the full advantage of our approach.
Another consequence of proposition 4.1 is worth pointing out: For every w belonging to W 2g 0 ,1 , let us define One has I (w) = I (w) .
Proof. In view of commutative diagram (16), Taking the image via P of both sides, and applying proposition 4.1 Considering the determinants, and recalling the explicit definition (7), The thesis follows from this last equation, in view of the fact that K 2g 0 ,1 is contained in GL 2g 0 , Z .
As a consequence, the quantity I is not constant over the whole W 2g 0 ,1 . In other words, it is not a trivial invariant with respect to the modular action M of the group G 2g 0 ,1 ⊂ Sp 4g 0 , Z .
To the best of our understanding, no geometrical interpretation of this quantity is available so far.

The case 2g 0 = 2
We start here to put in concrete action the previously developed abstract theory. The simplest case when 2g 0 equals 2 allows an elegant solution, together with a deeper understanding of the structure of the problem.

Lemma 5.1. One has
Moreover, if G belongs to G 2,1 , there are only two possible cases: either Proof. As pointed out in remark 3.5, one has the following inclusions: On the other side, This gives (42). Now, let G belong to G 2,1 . Then, either More explicitly, one has in the former case, and in the latter one. Always in view of remark 3.5, (45) and (46) yield (43) and (44) respectively.

Remark 5.2. The explicit characterization of the group G 2,1 given in the previous lemma can also be obtained by a direct calculation, starting from (2)
By means of (42) we are reconducted to a classical problem whose solution is well-known: the generic element of Sym >0 (2, R). A fundamental domain D ′′ for the congruent action of the group K 2,1 = GL (2, Z) on this space is described by the following system of inequalities This result was already known to Lagrange and Hermite.
Let us denote with For the map Σ defined in (14) one has the following explicit expression: Applying P to both sides of this equality, one gets These simple facts, together with the results of the previous sections, suffice to prove the following Theorem 5.3. A fundamental domain D for the modular action M of the group G 2,1 ⊂ Sp (4, Z) over W 2,1 is described by the following system of inequalities: Proof. Due to corollaries 4.3 and 2.2, such a fundamental domain D can be found by simply considering In view of (50), one gets an explicit description of this set operating the substitution into the system (48). This leads to (51).
With this result, the original problem can be considered completely solved when 2g 0 equals 2. The theory developed up to now, though, allows a deeper insight into the structure of the modular action of G 2,1 on W 2,1 .
Theorem 5.4. Let us introduce the following system of coordinates on the space W 2,1 : The square root is chosen to be positive. Consider the modular transformation given by an element of G 2,1 : In terms of the new coordinates, this acts as follows: and if G has the form (43), while Proof. The explicit expression for I (w) in (52) agrees with its general definition given in section 4. To see this it is sufficient to plug (50) into (41). Formula (54) is just the content of corollary 4.3. Gluing together (16) and (40) one gets the following commutative diagram: Let us introduce u (w) := 1 where the square root is again chosen to be positive. Using (57) one can write In view of definition (41), one has that det (u) = 1 ∀w ∈ W 2,1 .
This means that u (w) is not only symmetric and positive definite but also symplectic for every w belonging to W 2,1 . As a consequence of theorem 2.1, then, there exists a unique point τ (w) in the Siegel upper half plane of degree one such that Σ (τ (w)) = u (w) , w ∈ W 2,1 Using (14), (50) and (41) one can verify that this definition of τ coincides with the more explicit one given in (52). Now, suppose that G has the form (43). In this case, By point ii of theorem 2.1, (58) gives (55). Let G have the form (44), instead. One has that Equation (58) can be rewritten as follows: (59) By means of (14), one can verify that Again by point ii of theorem 2.1 equality (59) implies (56) Proposition 5.5. Consider the modular transformation associated to an element of G 2,1 , like the one in (53). One has sgn β = sgn β .
Proof. The explicit expression for I (w) given in (52) implies that On the other side, in view of (54) one has I (w) 1 ⇔ I (w) 1.

The general case
At the beginning of the last century Minkowski studied the congruent action of GL (n, Z) on Sym >0 (n, R): he exhibited, for every n ≥ 2, a fundamental domain M which can be described by a finite set of inequalities where f 1 , f 2 , . . . , f m n are linear homogeneous expressions of the entries of σ.
For low dimensions these expressions were explicitly determined: see [9] and [10]. After considerations of the previous sections, we now need a fundamental domain for the congruent action, on Sym >0 2g 0 , R , of the group K 2g 0 ,1 , which is stricly contained in GL 2g 0 , Z for 2g 0 ≥ 4. The index of this subgroup is finite. It can be easily computed and it equals 6 card GL 2g 0 , Z 2 card Sp 2g 0 , Z 2 (this is an advantage of the reduction procedure). In similar situations, a standard technique seems to be the following one: • Select a representantive for each left coset of K 2g 0 ,1 in GL 2g 0 , Z .
• Act with each representantive on the Minkowski fundamental domain M and consider the union of all the sets obtained this way.
• If its interior part is connected, this union is a fundamental domain for K 2g 0 ,1 .
As we are not aware of any reference for the theoretical justification of this procedure, we formulate one suitable for this setting here below Notation. Let g be an element of GL (n, Z). The symbol C g will denote the map defined as follows C g (σ) = C σ, g = g T σg, σ ∈ Sym >0 (n, R) . 6 Consider the quotient map φ : GL 2g 0 , Z −→ GL 2g 0 , Z 2 which associates to every unimodular matrix its class modulo 2 elementwise: it is surjective and (see [7], lemma 4 and [8] theorem 1 respectively); moreover Ker φ ⊂ K 2g 0 ,1 .
Because of these facts φ induces a bijection between the left cosets of K 2g 0 ,1 in GL 2g 0 , Z and the ones of Sp 2g 0 , Z 2 in GL 2g 0 , Z 2 . Lemma 6.1. Let σ be an interior point of M, and g ∈ GL (n, Z) such that C σ, g ∈ M.
Proof. By definition of fundamental domain, if C σ, g ∈ M then C σ, g = σ.
Now, letB σ be a neighborhood of σ contained in M. Due to the continuity of the map C g there exists a neighborhood of σ, sayB σ , such that Let us put B σ =B σ ∩ int (M). The points of B σ are all interior to M and remain inside M when the map C g is applied, so one has But B σ is an open subset of Sym >0 (n, R) and C g is a linear map, so This implies that g = ±Id.

Lemma 6.2. Let σ ∈ M. Every neighborhood of σ contains interior points of M.
Proof. This is a direct consequence of the fact that M is a convex set with non-empty interior. C M, g i for some g 1 , g 2 , . . . , g n ∈ GL (n, Z) and let g ∈ GL (n, Z) satisfy C ρ, g = σ for some ρ ∈ M. Then g = ±g i for some 1 ≤ i ≤ n.
Proof. Let B σ be a neighborhood of σ contained in n i=1 C M, g i , and let B ρ be a neighborhood of ρ such that (such a neighborhood exists due to continuity of the map C g ). Let ν be an interior point of M contained in B ρ ; then From this, applying lemma 6.1 one gets and fix a representative for each of them: for some g ∈ GL 2g 0 , Z . This relation can be rewritten as where C σ, g −1 1 belongs to M. Applying lemma 6.3 one gets that is g = ±g −1 1 g k for some 1 ≤ k ≤ m. Now, if k = 1 then g = ±Id and σ = ρ. If k 1, instead, g does not belong to K 2g 0 ,1 , because by construction g 1 and g k belong to different cosets. In the same way, one can treat the case in which σ belongs to C M, g j for j = 2, . . . , m and finally prove that ρ cannot be equivalent (in the K 2g 0 ,1 -sense) to any other point of m i=1 C M, g i .
At this point, the issue is about selecting a set of representatives such that the set in (61) is connected. Next lemma gives a working criterion to individuate some identifications on the border of M which will serve as "building blocks" in this task. Lemma 6.5. Let g ∈ GL (n, Z). Let us assume that there exists a point ρ 0 ∈ M with the following two properties: • There exists an integer k ρ such that f k ρ ρ 0 = 0 and f j ρ 0 > 0 for j k ρ .
There exists an open ball B ρ 0 with center in ρ 0 such that every ρ belonging to B ρ 0 satisfies f j ρ > 0, j k ρ .
If the radius of B ρ 0 is sufficiently small, its image through C g , say B σ 0 , has the analogous property: every σ belonging to B σ ′ satisfies Let B ρ 0 satisfy both properties above, and define the following three sets: Then C ρ, g belongs to the interior of M and is equivalent to another point of M, which is absurd. Suppose instead that f k σ C ρ, g < 0 and put ρ = ρ 0 + ν. Then ρ 0 − ν still belongs to B 0 ρ 0 and one easily proves that which again is absurd. It follows that This implies that there are only two possibilities: Since both B + ρ 0 and B + σ 0 are open sets contained in the interior of M, the first of (63) can occur only in the trivial case g = ±Id. In all other cases, then, B − σ 0 is contained in C M, g and The thesis now follows from (64) by means of standard arguments of general topology.
Suppose now that the matrices g 1 , g 2 , . . . , g l ∈ GL (n, Z) all satisfy the hypotheses of lemma 6.5. More generally one can consider finite collections of matrices of the form where 1 ≤ j 1 , j 2 , . . . , j k ≤ l. One can prove that also in this case the set int M ∪ C M, g j 1 ∪ C M, g j 2 g j 1 ∪ . . . ∪ C M, g j k g j k−1 . . . g j 1 is connected. It is possible to use this simple fact to try to determine algorithmically a set of matrices of GL (n, Z) which satisfies the hypotheses of proposition 6.4. We report an example here below.

6.1
The case 2g 0 = 4 Let us denote by the generic real, symmetric and positive definite matrix of dimension 4. A fundamental domain M for the congruent action of GL (4, Z) on Sym >0 (4, R) can be obtained by imposing the following conditions: ii for every the index i in the right-hand side of (66) depending on m as follows: This explicit result is due to E.S. Barnes and M.J. Cohn (see [9]). We used it to determine a fundamental domain D ′′ for the congruent action of K 4,1 on Sym >0 (4, R). Our calculations are summarized here below: The following elements of GL (4, Z) satisfy the hypotheses of lemma 6.5: This can be verified considering for each matrix above the corresponding point of M from the list below: where the matrices g j are listed in (68).

Discussion
The method presented here works, in principle, when 2g 0 is an arbitrary positive and even integer but it requires the explicit description of the Minkowski fundamental domain M. Such a description, though not completely non-redundant, is available for 2g 0 = 6 (see [10]); it would be interesting to try to work out the calculations in this case, with a more systematic use of a computer.
To the best of our knowledge, no explicit description of M is yet available when 2g 0 is equal or larger than 8.

Acknowledgements
The author wishes to thank Professor Boris Dubrovin for kindly supervising this work and Professor Tamara Grava for valuable discussions.