HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.


Introduction
It is well known that the category of complex algebraic curves is isomorphic to the category of compact Riemann surfaces. For a complex algebraic curve generated by the polynomial F (x, y) with complex coefficients, the corresponding compact Riemann surface P is obtained as the regularisation of the set of complex solutions of the equation F (x, y) = 0. If the Riemann surface P has a finite number of holes and punctures, we will say that the corresponding complex curve has holes and punctures.
Similarly the category of real algebraic curves is isomorphic to the category of pairs (P, τ ), where P is a compact Riemann surface and τ : P → P is an anti-holomorphic involution on P . For a real algebraic curve generated by the polynomial F (x, y) with real coefficients, the corresponding pair (P, τ ) consists of the compact Riemann surface P as above and the involution τ which is generated by the complex conjugation (x, y) → (x,ȳ). The set of fixed points of the involution τ is called the set of real points of the real curve (P, τ ) and is denoted by P τ . If the Riemann surface P has a finite number of holes and punctures, we will say that (P, τ ) is a real curve with holes and punctures. Those holes and punctures that are invariant under the involution τ will be called real holes and real punctures. The remaining holes and punctures occur in pairs which are mapped by the involution τ .
The set of real points of a compact real curve decomposes into pairwise disjoint simple closed smooth curves that are called ovals. The topological type of the compact real curve (P, τ ) is determined by the triple (g, k, ε), where g is the genus of P , k is the number of ovals and ε ∈ {0, 1} with ε = 1 if P \P τ is orientable and ε = 0 otherwise.
We can compactify a real curve (P, τ ) by closing every hole with a disk and every puncture with a point and extending τ to an anti-holomorphic involution τ on the resulting surface P . The ovals of ( P , τ ) are called compactified ovals. The type of a compactified oval is given by the cyclic sequence of glued-in disks and points. The topological type of the real curve (P, τ ) is determined by the type (g, k, ε) of the corresponding compactified real curve ( P , τ ), the numbers 2n I and 2m I of non-real holes and punctures and the types of compactified ovals. The numbers n R and m R of real holes and punctures respectively are determined by the type of the real curve. In this paper we will assume that 2g + 2n I + n R + 2m I + m R > 2 as the other cases are easier, but require different techniques.
The aim of this paper is to prove the following description of the moduli space of real curves with holes and punctures. is not empty if and only if 1 k g + 1 and k ≡ g + 1 (mod2) in the case ε = 1 and 0 k g in the case ε = 0. Under these conditions the moduli space M t has a natural topological structure, is connected and is the quotient T t /Mod t , where T t is homeomorphic to a real vector space of dimension 3g −3+3n I +2m I +2n R +m R and Mod t is a discrete group.
Acknowledgements. The acceptance of this paper for publication has been overshadowed by Sergey's death in December 2020. We completed and submitted the paper together except for the final edits to the references which I have done my best to complete. I would like to thank the referee for the helpful comments. I am grateful to Sergey for our productive collaboration and friendship over many years. (Anna Pratoussevitch in January 2021.)

Sequential sets of automorphisms
We will recall some standard results in complex analysis (see for example [5,Chap. 5] where λ > 0. A hyperbolic element with one fixed point at ∞ is of the form where α resp. β is the real fixed point and λ > 0. The parameter λ > 0 is called the shift parameter. The axis (g) of the element g = τ α,β (λ) is the geodesic between the fixed points α and β, oriented from β to α if λ > 1 and from α to β if λ < 1. The element g = τ α,β (λ) preserves the geodesic (g) and moves the points on this geodesic in the direction of the orientation. We call a hyperbolic element τ α,β (λ) with λ > 1 positive if α < β.
An element is called parabolic if it has one fixed point, which is on the boundary ∂H. A parabolic element with a fixed point α is of the form We call a parabolic element π α (λ) positive if λ > 0. An element that is neither hyperbolic nor parabolic is called elliptic. It has one fixed point that is in H. Given a base-point x ∈ H and a real number ϕ, let ρ x (ϕ) denote the rotation through angle ϕ about the point x. Any elliptic element is of the form ρ x (ϕ), where x is the fixed point.
We will call hyperbolic and parabolic automorphisms of H shifts. Riemann surfaces are bi-holomorphic to quotients H/Γ, where Γ is a Fuchsian group that consists of shifts. Sequential sets are special generating sets of such Fuchsian groups which were introduced in [1, p. 151]. They can be defined as follows: Definition 1. For two elements C 1 and C 2 in Aut(H) with finite fixed points in R we say that C 1 < C 2 if all fixed points of C 1 are to the left of any fixed point of C 2 .
and for some element A ∈ Aut(H) the elements have finite fixed points and satisfy C 1 < C 2 < C 3 . Figure 1 illustrates the position of the axes of the elements C i when all elements are hyperbolic.    . . , C r ) in Aut(H) is a sequential set if for any j ∈ {2, . . . , r − 1} the triple (C 1 · · · C j−1 , C j , C j+1 · · · C r ) is a sequential set.

Definition 4.
A sequential set of type (g, n, m) is a (n + m + 2g)-tuple of shifts (C 1 , . . . , C n+m , A 1 , B 1 , . . . , A g , B g ) in Aut(H) such that the elements A 1 , . . . , A g , B 1 , . . . , B g and C 1 , . . . , C n are hyperbolic, the elements C n+1 , . . . , C n+m are parabolic, and the tuple is a sequential set.   , a sequential set V of type (g, n, m) generates a Fuchsian group Γ(V) that consists of shifts such that the surface P = H/Γ is of genus g with n holes and m punctures. We will say that P is a surface of type (g, n, m).

Real surfaces without real holes or real punctures
Let the moduli space M g,n,m be the space of classes of bi-holomorphic equivalence of Riemann surfaces of type (g, n, m). According to [4,Thm. 2.1], any Riemann surface of type (g, n, m) is bi-holomorphic to a quotient H/Γ, where Γ is a Fuchsian group generated by a sequential set of type (g, n, m), hence we can describe the moduli space M g,n,m via the space of sequential sets of type (g, n, m). Let T g,n,m be the set of all sequential sets of type (g, n, m). The group Aut(H) ∼ = PSL(2, R) acts on the set T g,n,m by conjugation. Let T g,n,m = T g,n,m /PSL(2, R) be the quotient of this action. A sequential set of type (g, n, m) consists of 2g + n hyperbolic and m parabolic automorphisms which can be described by 3(2g + n) + 2m = 6g + 3n + 2m real parameters. The relation implies some restrictions on these parameters, hence a point in T g,n,m can be described by 6g + 3n + 2m − 3 real parameters. Taking into account the action of the 3-dimensional group PSL(2, R), we can conclude that the space T g,n,m is homeomorphic to R 6g+3n+2m−6 , see [4,Thm. 4.1].
Sequential sets in the same orbit of PSL(2, R) correspond to bi-holomorphically equivalent Riemann surfaces, hence the moduli space M g,n,m of Riemann surfaces of type (g, n, m) up to bi-holomorphic equivalence is of the form M g,n,m = T g,n,m /Mod g,n,m , where Mod g,n,m is a discrete group of automorphisms of a surface of type (g, n, m) that acts on T g,n,m . Therefore This fact is due to O. Teichmüller, see [7], §49 on p. 360 and §143 on p. 458 of the English translation. Now let us consider real algebraic curves. Recall that the topological type of a real curve without real holes and real punctures is determined by the type (g, k, ε) of the corresponding compactified real curve and the numbers 2n I and 2m I of non-real holes and punctures. A slight modification of the proofs in [4,Lem. 2.2] leads to the following description of the corresponding Fuchsian groups. A real curve of type t = (g, k, ε|2n I , 2m I ) can be constructed using a sequential set of type (h, g − 2h + 1 + n I , m I ) and hyperbolic automorphisms where the number h and the hyperbolic automorphisms D depend on the type t. The hyperbolic automorphisms D are constructed using reflections in geodesics in the hyperbolic space H. Let R j denote the reflection in the axis of the hyperbolic automorphism C j . For ε = 1, let h = (g − k + 1)/2 and D j = R 0 R j for j = 1, . . . , g − 2h = k − 1. For a hyperbolic automorphism C, let √ C be the hyperbolic automorphism such that ( Figure 3 shows the axes of the elements with the notation   We can show that automorphisms in V ∪ D generate a Fuchsian group Γ such that the Riemann surface P = H/Γ is of type (g, 2n I , 2m I ). Let σ = R 0 for k > 0 and σ = √ C 0 R 0 for k = 0, then σΓσ = Γ. It follows that σ generates a real curve (P, τ ) of type (g, k, ε|2n I , 2m I ). Moreover, it can be shown that every real curve of type t can be constructed in this way.
The hyperbolic metric on the upper half-plane H induces a hyperbolic metric on the quotient P = H/Γ. This metric reflects the geometry of the Fuchsian group Γ. For instance, among all simple closed curves around the hole that corresponds to the generator C j there is a unique shortest curve c j . This curve is the image of the axis of C j under the natural projection H → P .
Recall that real curves of type t = (g, k, ε|2n I , 2m I ) are generated by sequential sets of type (h, g − 2h + n I , m I ), whereby conjugate sequential sets correspond to bi-holomorphically equivalent real curves. The space T h,g−2h+1+n I ,m I of conjugacy classes of such sequential sets is homeomorphic to a real vector space of dimension 6h + 3(g − 2h + 1 + n I ) + 2m I − 6 = 3g − 3 + 3n I + 2m I .
The moduli space M (g,k,ε|2n I ,2m I ) is obtained as the quotient of T h,g−2h+1+n I ,m I by the discrete group Mod t of homotopy classes of those automorphisms that commute with τ . Thus M (g,k,ε|2n I ,2m I ) is homeomorphic to R 3g−3+3n I +2m I /Mod t .

Real surfaces of genus zero
We will now prove Theorem 1 for real surfaces of genus zero without non-real holes or punctures. In this case the compactified real curve has exactly one oval and the number of holes and punctures on this oval determines the type t of the real surface. Let us consider a set 1 , . . . , r , r > 2, of pairwise disjoint geodesics in the hyperbolic plane H such that the end points of i are to the left of the endpoints of i+1 . Let R i be the reflection in i . The product of two hyperbolic reflections is parabolic if their axes share exactly one endpoint and hyperbolic if the closures of their axes are disjoint. We can choose the geodesics i in such a way that the distribution of parabolic and hyperbolic elements among the products corresponds to the distribution of punctures and holes in the topological type t, see Figure 4.
Then (F 1 , C i , F 2 ) is a sequential set and hence so is (C 1 , . . . , C r ). Therefore the shifts C 1 , . . . , C r generate a Fuchsian group Γ and P r = H/Γ is a Riemann surface of genus zero. Moreover, R i ΓR i = Γ, hence the reflections R i induce an antiholomorphic involution τ : P r → P r such that all holes and punctures are real with respect to τ and their distribution corresponds to the distribution prescribed in the topological type t.
We will now prove that this construction gives all real curves (P r , τ ) of genus zero with only real holes and punctures. To this end, given such a real curve, we want to identify the corresponding geodesics 1 , . . . , r in H. Consider a connected component of P r \P τ r . Let Q r be its pre-image under the uniformisation map H → P r . The boundary of Q r consists of geodesic segments in H. We can extend these geodesic segments to geodesics i .
Sets of geodesics are determined by their endpoints. The number of endpoints is 2n + m, where n and m are the numbers of holes and punctures in the type t respectively. Taking into account the action of the 3-dimensional group PSL(2, R) of automorphisms of H on the set of sets of geodesics, we can conclude that the set of orbits of this action is homeomorphic to R 2n+m−3 , thus M t R 2n+m−3 /Mod, where Mod is a discrete group determined by the numbering of the holes.
We will now construct the space M * of real curves of genus zero with 3 holes of which exactly one is real. Let us revisit the previous construction with r = 3. We consider a set 1 , 2 , 3 of pairwise disjoint geodesics in the hyperbolic plane H such that the end points of i are to the left of the endpoints of i+1 . Let R i be the reflection in i . We can choose the geodesics i in such a way that all products C 3 = R 3 R 1 , C 1 = R 1 R 2 , C 2 = R 2 R 3 are hyperbolic. Let c i be the axis of C i for i = 1, 2, 3. In this case the geodesics 1 , 2 , 3 and c 1 , c 2 , c 3 bound a right-angled hexagon Q 3 in H. The type of the hexagon Q 3 is determined up to bi-holomorphic equivalence by the lengths of its sides that are contained in the geodesics c 1 , c 2 , c 3 , see Theorem 3.5.13 in [6]. These lengths are given by (λ 1 /2, λ 2 /2, λ 3 /2), where λ i = λ(C i ) is the shift parameter of the hyperbolic element C i . They are completely determined by the lengths of the minimal geodesics around the holes of the surface P 3 . If λ 1 = λ 2 then the hexagon Q 3 is symmetric with respect to the geodesic orthogonal to c 3 , see Figure 5. The reflection in this geodesic generates an antiholomorphic involution σ : P 3 → P 3 that interchanges the holes c 1 and c 2 and maps the hole c 3 to itself. Note that λ 1 and λ 3 completely determine the real curve (P 3 , σ) and hence the space M * of such real curves is homeomorphic to R 2 .   Now let us construct the space M * s of real curves (P, τ ) of genus zero with two non-real holes and r real holes and punctures, generating an oval of type s. Consider the minimal closed geodesic c 0 that separates non-real holes from real holes and punctures. The geodesic c 0 divides the real curve (P, τ ) into two real curves (P r+1 , τ ) and (P 3 , σ) of the types considered above. On the other hand, starting with the real curves (P r+1 , τ ) and (P 3 , σ), we can glue them together to form a real curve of type t if the length of the minimal closed geodesic around the real hole on (P 3 , σ) is equal to the length of the minimal closed geodesic around the hole C 1 on (P r+1 , τ ). Therefore M t R 2n R +m R −3 /Mod, where n R and m R are the numbers of real holes and real punctures in the type s respectively.

Real surfaces of any type
We will now consider the space of real surfaces of any type. Such real surfaces can be obtained by identifying pairs of non-real symmetric holes on the kinds of surfaces considered in Sections 2 and 3.
Consider a real curve (P, τ ) of type t without real holes or real punctures. Let c be a minimal closed geodesic around a non-real hole on (P, τ ). Let ρ be the length of c. Let c 0 be as described in Section 2. Among all geodesic segments connecting the curves c and c 0 there is a unique shortest segment. Let x be the end point of this segment on the curve c.
Consider a real curve (Q, σ) of type s with exactly two non-real holes and without non-real punctures. Let u and σ(u) be the minimal simple closed geodesics around the non-real holes on (Q, σ). We can choose (Q, σ) in such a way that the length of u and σ(u) is equal to ρ. Let c 0 be the minimal closed geodesic on (Q, σ) that separates non-real holes from real holes and real punctures. Among all geodesic segments connecting the curves u and c 0 there is a unique shortest segment. Let y be the end point of this segment on the curve u.
The family of isometries ϕ mapping the geodesic c to the geodesic u is real one-dimensional and can be parametrised by the distance between the points ϕ(x) and y. We can identify the holes τ (c) and σ(u) via the isometry σ • ϕ • τ . Thus we can glue a real curve (Q, σ) of genus zero and type s into a non-real hole of a real surface (P, τ ) of type t. Repeatedly gluing in real surfaces of genus zero we can obtain a real curve of any type.
Consider the space M (t|s1,...,sr) of real curves of topological type (t|s 1 , . . . , s r ), where t is the topological type of a real curve without holes and punctures and s i are the types of ovals. The results of Sections 3 and 4 imply that M (t|s1,...,sr) = M t × R 2n R +m R .
Recall that we have shown in Section 2 that M t R 3g−3+3n I +2m I /Mod t , where t = (g, k, ε|2n I , 2m I ).