Algebraic intersection for hyperbolic surfaces
Manman Jiang Huiping Pan


Inside the echo chamber: Linguistic underpinnings of misinformation on Twitter
    Sarah Rajtmajer
    April 29, 2024
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We show that the algebraic intersection form of hyperbolic surfaces of genus g has a minimum in the moduli space and that the minimum grows in the order (log g)^-2  in terms of the genus.  We also describe the asymptotic behavior of the algebraic intersection form in the moduli space as the homologically systolic length goes to zero.


MSC classification: 30F60, 32G15, 30F45




§ INTRODUCTION

 





 §.§ Main results

Given a Riemannian surface X, the algebraic intersection form K(X) is defined to be: 

    K(X)=sup_α,β|(α,β)|/ℓ_X(α)ℓ_X(β),

where (·,·) represents the algebraic intersection number,
where α and β range over all oriented simple closed geodesics under the Riemannian metric X, and where ℓ_X(· ) represents the length of geodesics with respect to X. The algebraic intersection form K(X) and its volume (X) of a Riemannian surface X satisfies K(X)(X)≥ 1 with equality holds if and only if X is a flat torus <cit.>. 
     
     For hyperbolic surfaces, Massart and Muetzel <cit.> proved that K(X) is unbounded and nonproper in the moduli space _g of hyperbolic surfaces of genus g≥2.  In particular, K(X) does not have a maximum in _g. Regarding the minimum of K(X), they asked:
     [<cit.>, Question 1.7]
     	Does K(X) have a minimum in the moduli space _g? If so, which surface realizes the minimum? 
     
   
In this paper, we answer the first part of the above question affirmatively:
For g≥2, the algebraic intersection form K(X) attains its   minimum in the moduli space _g. 


Next, we estimate the minimum in terms of the genus:
  Let g≥2. 
There exist  two universal constants C_1, C_2 such that 
	
    C_1/(log g)^2≤min_X∈_gK(X)≤C_2/(log g)^2.
 


In <cit.>, Boulanger also discussed the minimum, or more precisely the inifimum, and sketched a proof of the upper bound. The upper bound also follows directly from <cit.>. Our main contribution here is the lower bound.


This leads to the question:

	Does the limit lim_g→∞ (log g)^2 min_X∈_g K(X) exist as g→∞?

	
	Another interesting question is to study the asymptotic behavior in the moduli space. Massart and Muetzel <cit.> observed that it is possible to pinch a homologically trivial simple closed geodesic while keeping the algebraic intersection form bounded from above. On the other hand, pinching a homologically nontrivial simple closed geodesic would alway make the algebraic intersection form diverge to infinity <cit.>:
	
		There exist constants A and B depending only on the genus g such that for any X∈_g with the homologically systolic length _h(X) sufficiently small,  we have
		
    A/_h(X)· |log_h(X)|≤ K(X)≤B/_h(X)· |log_h(X)|.

	
	
	
	Here the homologically systolic length _h(X) is the length of any shortest homologically nontrivial simple closed geodesic on X. Based on Theorem <ref>, it is natural to wonder if it is possible to describe the asymptotic behavior in terms of homologically systolic length. However, this will not work. For any g≥2 and for any 1≤ c≤ 2,  we construct a sequence of hyperbolic surfaces {X_n} in _g such that, as n→∞, we have 
	
    K(X_n)∼1/2c·_h(X_n)|log_h(X_n)|.

	 Here A∼ B means lim A/B=1. See Section <ref> for the construction.  This example demonstrates that the asymptotic behavior of the algebraic intersection form cannot be described as a single function of the homologically systolic length. Instead, all short curves should be taken into consideration. 


 For g≥2 and X∈_g, let Γ be the set of nonseparating simple closed geodesics of length at most 1. For each γ∈Γ, let α_γ be one of the shortest nonseparating simple closed geodesics that intersect γ exactly once. Then  as _h(X)→0, 
      
    K(X)∼max_γ∈Γ1/ℓ_X(γ)ℓ_X(α_γ).

		
		Here the constant 1 in Theorem <ref> can be replaced by any positive constant. The set Γ may contain curves that are disjoint from any homological systole and cannot be replaced by the set of curves intersecting homological systoles (see Example <ref> in Section <ref>).
	
	


 §.§ Background and motivation

Motivated by the question of how much can two curves of a given length intersect, Massart and Muetzel <cit.> initiated the study of algebraic intersection form of surfaces with Riemannian metrics.  The algebraic intersection form plays an important role in the comparison between the stable norm and the Hodge norm (L^2-norm) <cit.>.  In the same paper, Massart and Muetzel also estimated the algebraic intersection form in terms of the homologically systolic length and the diameter.

	 Recently, the algebraic intersection form of translation surfaces has attracted much attention.  Cheboui,  Kessi, and Massart <cit.> considered the scaling invariant quantity KVol(X):=K(X)(X) in the stratum ℋ(2) of translation surfaces of genus two with a single cone point.  They computed the exact values of KVol for some family of square-tiled surfaces, proved that  inf_X∈ℋ(2)KVol(X)≤ 2 and constructed a sequence of square-tiled surfaces  X_n in ℋ(2)  such that KVol(X_n)→ 2 with 2 being the conjectural infimum for KVol over ℋ(2). Boulanger <cit.> generalized this picture to the minimal stratum ℋ(2g-2) of genus g with a single cone point, proving that the infimum of KVol over each component of ℋ(2g-2) is at most g and constructing surfaces with KVol converging to g with g being the conjectural infimum. Beyond square-tiled surfaces, Boulanger, Lanner and Massart computed exact values of KVol for some non-arithmetic Veech surfaces,  the regular double n-gon translation surfaces for n≥5 odd <cit.> and regular n-gon translation surfaces for n≥8 even <cit.>.
	 
	 	
Using geometric intersection number ı(·,·), Torkaman <cit.> initiated the study of the intersection strength I(X),  defined by 
 
    I(X)=sup_α,βı(α,β)/ℓ_X(α)ℓ_X(β)
 
 where the supremum ranges over all closed geodesics on X. She proved that I(X) is a proper function on the moduli space _g,  described the asymptotic behavior in terms of the systolic length (X) as X approaches the boundary of _g,
 
    I(X)∼1/2·(X)|log(X)|,

  and showed that the minimum of I(X) in the moduli space _g also grows in the order (log g)^-2 (by showing that every hyperbolic surface of genus g has a figure eight closed geodesic of length at most clog g for some universal constant c).  From the definition of K(X) and I(X), we see that K(X)≤ I(X), and hence min_X∈_gK(X)≤min_X∈_g I(X). It is natural to ask whether they are equal:
 
 	Are min_X∈_g K(X) and min_X∈_g I(X) equal to each other? 
 

Another closely related quantity is the systolic volume/area SysVol(X) defined by:
 
    SysVol(X):=(X)/(X)^2,
 
 where (X) is the systolic length of X, that is, the length of any homotopically nontrivial shortest simple closed geodesic on X. For recent results about this quantity, we refer to  <cit.>. 
  
  

 §.§ Sketch of the proof
 The possible obstruction for the existence of a minimum of the algebraic intersection form in the moduli space is the existence of surfaces with short and homologically trivial simple closed geodesics.  We begin with an observation that, for the surfaces mentioned above, the algebraic intersection form is the maximum among that of the complementary components of short curves (Lemma <ref>). We then apply the strip deformation to lengthen those short curves and simple closed geodesics disjoint from them (Lemma <ref>), pushing the original surface into a compact subset of the moduli space and decreasing the algebraic intersection form. This ensures the existence of a minimum.  
  
     As mentioned earlier, the upper bound of Theorem <ref> is already known in literature <cit.>. For the lower bound, the estimate relies crucially on the existence of short homological bases <cit.> if the homological systole of the underlying surface is bounded from below. This allows us to reduce the problem to considering surfaces with short but not too short homological systoles. We then apply the Fenchel-Nielsen coordinates to lengthen those short curves to some uniform constant while keeping other simple closed geodesics from being too contracted (Lemma <ref>), yielding the desired lower bound.   
     
     For the asymptotic behavior in the moduli space, the key is to find good representatives within the homology classes of simple closed geodesics:  crossing each short curve at most once (Lemma <ref>) and having small twists around each short curve ( Lemma <ref>).
     
      

 §.§ Acknowledgement
 The first author is supported by the National Natural Science Foundation of China NSFC 12371076. The Second author is supported by National Natural Science Foundation of China NSFC 12371073 and Guangzhou Basic and Applied Basic Research Foundation 2024A04J3636.
 
 







§ STRIP DEFORMATIONS



 §.§ Teichmüller space

Let S be a closed orientable surface of genus g≥2. The space _g of S is the equivalence classes of hyperbolic metrics on S, where two metrics are said to be equivalent if there exists an isometry between them which is homotopic to the identity map. The space is topologically homeomorphic to ^6g-6. The moduli space _g is also defined as the equivalence classes of hyperbolic metrics on S, where two metrics are said to be equivalent if there exists an isometry between them. The moduli space is the quotient space of the space under the mapping class group.

 Given a hyperbolic surface X and a simple closed curve α, we denote by ℓ_X(α) the length of the unique geodesic homotopic to α. The shortest homotopically nontrivial closed geodesics on X are called systoles of X, and the shortest homologically nontrivial closed geodesics are called homological systoles of X. We denote by (X) and _h(X)  respectively the lengths of systoles and homologically systoles of X. 
     
The  geometric intersection number ı(α,β) of two closed curves α and β on S is defined to be  the minimum of the cardinality of α'∩β' where α' and β' range over all curves homotopic to α and β, respectively.
 
 Given   a volume form ω on S, he  algebraic intersection number (α,β) of two differentiable oriented closed curves α:[0,1]→ S and β:[0,1]→ S is defined as follows:
 
    (α,β):=∑ω(α̇(s),β̇(t))/|ω(α̇(s),β̇(t))|,
 
 where α̇ and β̇ represent the tangent vector of α and β respectively and where the sum ranges over all pairs (s,t) such that α(s)=β(t).
 
For more information about hyperbolic surfaces, Teichmüller space, and moduli space, we refer to <cit.>.



 §.§ Strip deformations

In this section, we introduce the
 strip deformations, first introduced by Thurston (<cit.>, see also <cit.> and <cit.>) 

A strip of width w≥0 is a region on ^̋2 bounded by two complete geodesic lines whose distance is exactly w. The two closest points on distinct components of the strip are called the feet of the strip. 

Let Y be a complete hyperbolic surface of finite genus and infinite area whose convex hull (Y) is a boarded hyperbolic surface. Let η={η_i} be a collection of pairwise disjoint simple, proper geodesic arcs. Let P={P_i} be a collection of marked points with P_i∈η_i. Let w={w_i} be a collection of non-negative real numbers. The  strip deformation of Y determined by the triple (η,P,w) is a new complete hyperbolic surface Y' obtained by cutting along each η_i  and gluing in a strip of width w_i so that the two points on the boundary of Y\η_i  corresponding to the marked point P_i∈η_i are identified with the feet of the strip. 

The length of every closed geodesic on Y' is bigger than the corresponding length on Y, and strictly bigger if the geodesic in consideration intersects η (see  <cit.> for instance). Furthermore, if the weight w_i∈w goes to infinity, then the length of every closed geodesic intersecting the arc η_i also goes to infinity. 

 Recall that every boarded hyperbolic surface is the convex hull of a unique complete hyperbolic surface. Whenever we say a strip deformation of a boarded hyperbolic surface Y_0, we mean the convex hull of the corresponding strip deformation of the complete hyperbolic surface which contains Y_0 as its convex hull. 

	In the remainder of the paper, we make the following convention. 
	
		
  
  	We will also call a homologically trivial simple closed geodesic a separating simple closed geodesic, and call a homologically nontrivial simple closed geodesic a nonseparating simple closed geodesic. 
  
   
	

	Let X∈_g. Let {γ_1,⋯,γ_k} be a collection of disjoint simple closed geodesics. Let δ be a constant such that δ≥ℓ_X(γ_i). Then there exists a hyperbolic surface X'∈_g such that 
	
		
  * ℓ_X'(γ_i)=δ for each i, and 
		
  * ℓ_X'(α)≥ℓ_X(α) for every simple closed geodesic α on X\(γ_1∪⋯∪γ_k). 
	


	Let X_1,⋯,X_m be the components of X\(γ_1∪⋯γ_k). Applying strip deformations  to each X_i, we see that there exists a boarded hyperbolic surface X_i' homeomorphic to X_i such that
		
		
  * the length of each geodesic boundary component of X_i is exactly δ,
		
  * the length of every simple closed geodesic α in the interior of X_i satisfes: ℓ_X_i'(α)≥ℓ_X_i(α).
	
	Let X' be a hyperbolic surface obtained by gluing X_1',X_2',⋯,X_m' along pairs of geodesic boundary components corresponding to γ_i for all i. (The gluing process is not unique. The resulting surfaces differ by a twist about {γ_1,⋯,γ_k}.)  Then ℓ_X'(α)≥ℓ_X(α) for every simple closed geodesic α on X\(γ_1∪⋯∪γ_k) .  


There are other approaches to lengthen simple closed geodesics on boarded hyperbolic surfaces: using Fenchel-Nielsen coordinates (see 
	 <cit.> for instance, as well as the following section), taking the Nielsen kernel of a boarded surface <cit.>,  or more generally, using  pruning constructions <cit.> (the inverse construction of grafting).  One advantage of strip deformation is that we could uniformly lengthen all closed geodesics if the set α of arcs cuts the underlying hyperbolic surface into topological disks.




§ THICK-THIN DECOMPOSITION AND FENCHEL-NIELSEN COORDINATES


The goal of this section is to use the Fenchel-Nielsen coordinates to lengthen short curves to some uniform constant while keeping other simple closed geodesics from being too contracted (Lemma <ref>). This will be used later for estimating the lower bound of the minimum of the algebraic intersection form in the moduli space. 


 §.§ Collar neighbourhoods and thick-thin decomposition of hyperbolic surfaces

Given a hyperbolic surface X∈_g and a simple closed geodesic γ, let (γ) be the collar neighborhood of γ defined by:		
    (γ):={p∈ X:dist(p,γ)≤arcsinh1/sinh (ℓ_X(γ_i)/2)}.

		Each circle inside (γ) of distance ρ from γ has length 
		
    ℓ_X(γ)coshρ≤ℓ_X(γ) √(1+1/sinh^2 ℓ_X(γ)/2)≤ 2 √(1+sinh^2 (ℓ_X(γ)/2)).

		In particular, if  ℓ_X(γ)≤ 2arcsinh1, then each of the boundary components of (γ) has length at most 2√(2). Note that (γ) is also foliated by geodesic arcs, each of which connects one boundary component to the other and intersects γ perpendicularly. We call any such geodesic arc a perpendicular of (γ). Let ξ be a geodesic arc from one boundary component of (γ) to the other component. Suppose that ξ crosses some perpendicular  m times. Then it crosses any other perpendicular at most m+1 times. By an elementary calculation (<cit.>), we see that, if ℓ_X(γ)≤ 2arcsinh1, then
	
    |ℓ_X(ξ)-(2|logℓ_X(γ)|+mℓ_X(γ))|≤ C
 
	for some universal constant C. 
	
		Hyperbolic surfaces with short curves admit  thick-thin decompositions.
	Let {γ_1,⋯,
	γ_k} be the set of simple closed geodesics on X of length at most 2arcsinh1. Then (i) γ_1,⋯,γ_k are pairwise disjoint, and (ii) any point outside ∪_i (γ_i) has injective radius at least arcsinh  1≈0.88 <cit.>. The union ∪_i (γ_i) is called the thin part of X and its complementary is called the thick part of X.  
	
	We end up this subsection with the following convention: 
	
		
		
			Fixed a constant 0<<arcsinh1≈0.88 once for all such that for any 0<x<, 
	
    arcsinh1/sinh (x/2)≥ |log x|≥√(2).

		We may choose =e^-√(2)≈0.24.
		
	
	
				 




 §.§ Fenchel-Nielsen coordinates


In this subsection, we introduce the Fenchel-Nielsen coordinates on the Teichmüller space _g.  Let Γ  be a pants decomposition of S_g whose complementary S\Γ is a union of pairs of pants. 



For γ∈Γ, the twist of X∈_g with respect to Γ at γ is defined in the following way. Let R' and R” be the pairs of pants in S\Γ, each of which contains γ as a boundary component. Let γ' (resp. γ”) be a geodesic boundary component of R' (resp. R”) different from γ. Let η' be the unique simple geodesic arc on R' connecting γ' and γ perpendicularly. Let η” be similarly defined. Choose an orientation of γ such that R' sits to the left of γ.  Let d_γ(X)∈ [-ℓ_X(γ)/2,ℓ_X(γ)/2) be the oriented distance from the foot η'∩γ to the foot η”∩γ along γ with respect to prescribed orientation (see Figure <ref>). (In fact, the signed distance is independent of the choice of the orientation of γ. Instead, it depends on the orientation of the underlying surface.)   Let  

    t_γ(X):=d_γ(X)/ℓ_X(γ)∈ [-1/2,1/2).

The function t_γ defines a function from _g to the unit circle /.  Accordingly, the function t_Γ:=(t_γ)_γ∈Γ defines a map from the Teichmüller space to the torus 𝕋^3g-3:=^3g-3/^3g-3. Let X_0∈_g be a hyperbolic surface such that t_Γ(X_0)=0. Let t_Γ:={t_γ}_γ∈Γ: ^3g-3→^3g-3 be the lift of t_Γ:^3g-3→^3g-3/^3g-3 such that t_Γ(X_0)=0.

The (6g-6)-tuple (ℓ_X(γ),t_γ(X))_γ∈Γ defines a Fenchel-Nielsen coordinates on the Teichmüller space _g.




   As we can see from the construction above, the Fenchel-Nielsen coordinate is not unique. The coordinate depends on the choice of Γ, the choice of η' and η” (which in turn is determined by the pairs {γ',γ} and {γ”,γ}) for each γ∈Γ, and the choice of X_0. 
       
      
      

 §.§ Twists of proper arcs on pairs of pants

      
      
      
      To estimate the length of simple closed geodesics in terms of Fenchel-Nielsen coordinates, we also need the twist of proper simple arcs on pairs of pants with respect to the corresponding pants curves. More precisely, let P be a pair of pants of X\Γ with boundary components, say  γ_1,γ_2, and γ_3. 
      
      Let ξ_12 be a simple arc on P which connects γ_i_1 to γ_i_2.  Let η_12 be the unique geodesic homotopic to ξ_12 relative to γ_i_1 and γ_i_2, that is, the unique simple geodesic arc which connects γ_i_1 and γ_i_2 perpendicularly (Figure <ref> (a)).  Now, consider the lift of P to the hyperbolic plane. Choose one lift η̃_12 of η_12 and let γ̃_i_1, γ̃_i_2, and ξ̃_12 be the corresponding lifts of γ_i_1, γ_i_2 and ξ_12. Let A_j (resp. B_j) be the intersection point between γ̃_i_j and η̃_12 (resp. ξ̃_12). Choose an orientation for  γ_i_j such that P sits to left  of  γ_i_j.  This induces an orientation on γ̃_i_j.  (see Figure <ref> (b)).  Let ρ_j(ξ_12)  be the oriented distance from A_j to B_j. The twist t_j(ξ_12) of ξ_12 at γ_i_j is defined to be:
      
    t_j(ξ_12):=ρ_j(ξ_12)/ℓ_X(γ_i_j).

      
      By the hyperbolic trigonometry formula (<cit.>), we see that the length ℓ_X(ξ_12) of the unique geodesic representative on P which is homotopic to ξ_12 relative to the endpoints satisfies:
      
    coshℓ_X(ξ_12)
       =   coshρ_1(ξ)coshρ_2(ξ)coshℓ_X(η_12)-sinhρ_1(ξ)sinhρ_2(ξ).
  
           About the length ℓ_X(η_12) of η_12, if γ_i_1≠γ_i_2, then, by <cit.>,  
      	
    coshℓ_X(η_12)=coshℓ_X(γ_i_3)/2+coshℓ_X(γ_i_1)/2coshℓ_X(γ_i_2)/2/sinhℓ_X(γ_i_1)/2sinhℓ_X(γ_i_2)/2,

      	where γ_i_3 is the third  boundary component of P. 	
If γ_i_1= γ_i_2, let γ_i_3 be one of the other two boundary components of P and let η_13 be the unique simple geodesic arc connecting γ_i_1 and γ_i_3 perpendicularly. By   <cit.> we see that 
      	
    coshℓ_X(η_12)/2=sinhℓ_X(η_13)sinhℓ_X(γ_i_3)/2.

   
      
   



 §.§ Surfaces with short curves

Given a hyperbolic surface X∈_g with systolic length (X)< <ref>, we shall construct a new hyperbolic surface with big systole and controlled geometry. More precisely, let γ_1,⋯,γ_k be the set of simple closed geodesics on X of length strictly less than <ref>. In particular, they are pairwise disjoint <cit.>.  We extend γ_1,⋯,γ_k to a pants decomposition Γ:={γ_1,⋯,γ_k,γ_k+1⋯,γ_3g-3}. Let (ℓ_X(γ),t_γ(X))_γ∈Γ be the Fenchel-Nielsen coordinates defined in Section <ref>. 

Let Y∈_g be the hyperbolic surface with Fenchel-Nielsen coordinates:
   
    (ϵ_M,⋯,ϵ_M,ℓ_X(γ_k+1),⋯, ℓ_X(γ_3g-3),t_γ_1(X),t_γ_2(X),⋯,t_γ_3g-3(X) ).

In other words, we replace the length of each of γ_1,⋯,γ_k by <ref> while keeping the other terms unchanged.




	Let X∈_g be a hyperbolic with (X)<<ref>. Let Y be defined as above.  Then (Y)=<ref>.  Furthermore, for every simple closed geodesic α on X, the following holds.
	
		
  * If α intersects the some γ_i,  then
			
    ℓ_X(α)/ℓ_Y(α)≤<ref>·  |log (X)|

	for some universal constant .
	 
  * If α is disjoint from any of γ_i, then 
	 	
    ℓ_X(α)/ℓ_Y(α)≤ 1.

	
	


  

Let α be an arbitrary simple closed curve on X. Let α^Y be the geodesic on Y which is homotopic to α. The (geodesic representative of the) pants decomposition Γ cuts α^Y into a collection {ξ_i^Y} of geodesic segments.  From the construction of Fenchel-Nielson coordinates in Section <ref>, we see that α^Y (not the homotopy class of α^Y) remains to be a simple closed curve on X, which is not necessarily a geodesic. To bound the length  ℓ_X(α) of its geodesic representative on X from above, we shall estimate the length Length_X(ξ^X_i) of the geodesic ξ_i^X on X which is homotopic to ξ_i^Y relative to its endpoints:

    ℓ_X(α)≤Length_X(α^X)≤∑_i Length_X(ξ_i^X).

To prove the lemma, it suffices to prove the following claim:


	
	Claim: Let ξ_i^X and ξ_i^Y be as above.
		
		
  * If ξ_i^X is disjoint from any of the short curves γ_1,⋯,γ_k, then 
    Length_X(ξ_i^X)/Length_Y(ξ_i^Y)≤ 1.
 
	
  * If ξ_i^X intersects at least one of the short curves γ_1,⋯,γ_k, then 	
    Length_X(ξ_i^X)/Length_Y(ξ_i^Y)≤<ref>· |log(X)|
 
	for some universal constant <ref>.
		
	

 Indeed, if α does not intersect any of the short curves γ_1,⋯,γ_k, then by the first item of the claim, we see that
 
    ℓ_X(α)≤Length_X(α^X)≤∑_i Length_X(ξ_i^X)≤∑_i Length_Y(ξ_i^Y)=ℓ_Y(α).

If α intersects at least one of the short curves γ_1,⋯,γ_k, then by the second item of the claim, we see that
 
    ℓ_X(α)   ≤   Length_X(α^X)≤∑_i Length_X(ξ_i^X)
       ≤   <ref>|log(X)|·∑_i Length_Y(ξ_i^Y)=<ref>|log(X)|·ℓ_Y(α).

It remains to consider the systolic length (Y). Let α be simple closed geodesic on Y. If α is one of γ_i, then ℓ_X(γ_i)=<ref>. If α is disjoint from any of γ_i, then the first item in the claim implies that ℓ_Y(α)≥ℓ_X(α)≥<ref>. If α intersects some γ_i. then it crosses the collar neighborhood (γ_i) of γ_i, which is of width 
arcsinh(1/sinh(ℓ_X(γ)/2))≥<ref> due to the assumption that ℓ_X(γ_i)<<ref>. Hence, ℓ_Y(α)≥ 2<ref>. This proves that (Y)≥<ref>. That ℓ_X(γ_i)=<ref> for i≤ k implies that (Y)=<ref>.



The remainder of the proof is devoted to proving the claim. To simplify notations, we set ℓ_X(ξ^X_i):=Length_X(ξ^X_i) and ℓ_Y(ξ^Y_i):=Length_Y(ξ^Y_i).  Let P be the pair of pants which contains ξ_i^X. Let γ_i_1 and γ_i_2 be two pants curves connected by the arc ξ^X_i (and hence also by ξ^Y_i). Let t_1(ξ) (resp. t_2(ξ)) be the twist of ξ^X at γ_i_1 (resp. γ_i_2) with respect to  the pair of pants P (see Section <ref> for relevant definitions). By the definition of ξ^Y, we see that t_1(ξ) (resp. t_2(ξ)) is also the twist of ξ^Y at γ_i_1 (resp. γ_i_2) with respect to  the pair of pants P.
Let η_i^X be the geodesic segment under the metric X  which is homotopic to ξ_i^X  relative to γ_i_1 and γ_i_2.   Let η_i^Y be the geodesic segment under the metric Y  which is homotopic to ξ_i^Y  relative to γ_i_1 and γ_i_2. There are two case depending on whether γ_i_1=γ_i_2 or not.

Case 1. γ_i_1≠γ_i_2. Let γ_i_3 be the third edge of the pair of pants P containing ξ^X. There are two more subcases depending on whether γ_i_1 or γ_i_2 is short.

Subcase 1-1:  neither γ_i_1 nor γ_i_2 is short, that is ℓ_X(γ_i_1)≥<ref> and ℓ_X(γ_i_2)≥<ref>. Then, by the construction of Y,

    ℓ_X(γ_i_1)= ℓ_Y(γ_i_1)  and ℓ_X(γ_i_2)= ℓ_Y(γ_i_2).

Since ℓ_X(γ_i3)≤ℓ_Y(γ_i3), it follows from (<ref>) that ℓ_X(η_i^X)≤ℓ_Y(η_i^Y). Hence, 
by (<ref>), we see that 
ℓ_X(ξ^X_i)≤ℓ_Y(ξ^Y_i). 

Subcases 1-2: either γ_i_1 or  γ_i_2 is short, say ℓ_X(γ_i_1)<<ref>. Accordingly, we have ℓ_Y(γ_i_1)=<ref>. By (<ref>), we see that 

    ℓ_X(η_i^X)   ≥   logcoshℓ_X(γ_i_1)/2coshℓ_X(γ_i_2)/2/sinhℓ_X(γ_i_1)/2sinhℓ_X(γ_i_2)/2
       ≥   logcoshℓ_X(γ_i_1)/2/sinhℓ_X(γ_i_1)/2≥logcosh<ref>/2/sinh<ref>/2,

and similarly, 

    ℓ_Y(η_i^Y)≥logcosh<ref>/2/sinh<ref>/2.

Again by (<ref>), we have 

    coshℓ_X(η_i^X)/coshℓ_Y(η_i^Y)
       =   coshℓ_X(γ_i_3)/2+coshℓ_X(γ_i_1)/2coshℓ_X(γ_i_2)/2/coshℓ_Y(γ_i_3)/2+coshℓ_Y(γ_i_1)/2coshℓ_Y(γ_i_2)/2·sinhℓ_Y(γ_i_1)/2sinhℓ_Y(γ_i_2)/2/sinhℓ_X(γ_i_1)/2sinhℓ_X(γ_i_2)/2
       ≤   sinhℓ_Y(γ_i_1)/2sinhℓ_Y(γ_i_2)/2/sinhℓ_X(γ_i_1)/2sinhℓ_X(γ_i_2)/2≤sinh ^2<ref>/2/sinh^2(X)/2

where in the last equality we use the assumption that ℓ_X(γ_i_1)<<ref>=ℓ_Y(γ_i_1) and the fact  that ℓ_Y(γ_i_2)=ℓ_X(γ_i_2) if ℓ_X(γ_i_2)≥<ref> or ℓ_Y(γ_i_2)=<ref> if  ℓ_X(γ_i_2)<<ref>.
 Hence

    ℓ_X(η_i^X)-ℓ_Y(η_i^Y)   ≤   log2+logcoshℓ_X(η_i^X)/coshℓ_Y(η_i^Y)≤log 2+2logsinh<ref>/2/sinh(X)/2.

Together with (<ref>), this implies that

    ℓ_X(η_i^X)/ℓ_Y(η_i^Y)   =    1+ℓ_X(η_i^X)-ℓ_Y(η_i^Y)/ℓ_Y(η_i^Y)
       ≤   <ref>· |log(X)|

for some universal positive constant .
Combining  (<ref>), (<ref>), (<ref>), and (<ref>), we see that there exist two universal constants  
 and   such that 
    ℓ_X(ξ_i^X)   ≤   <ref>· ( t_i_1(ξ_i)ℓ_X(γ_i_1)+ t_i_2(ξ)ℓ_X(γ_i_2)+ ℓ_X(η_i^X))

and

    ℓ_Y(ξ_i^Y)
    	    ≥   <ref>· (t_i_1(ξ_i)ℓ_Y(γ_i_1)+t_i_2(ξ)ℓ_Y(γ_i_2)+ℓ_Y(η_i^Y )).

Therefore,

    ℓ_X(ξ^X_i)/ℓ_Y(ξ^Y_i)   ≤   <ref>/<ref>·t_i_1(ξ_i)ℓ_X(γ_i_1)+ t_i_2(ξ)ℓ_X(γ_i_2)+ ℓ_X(η_i^X)/t_i_1(ξ_i)ℓ_Y(γ_i_1)+ t_i_2(ξ)ℓ_Y(γ_i_2)+ ℓ_Y(η_i^Y)
       ≤   <ref><ref>/<ref>· |log(X)|.          ( by (<ref>))


Case 2. γ_i_1=γ_i_2.  Let γ_i_3 be one of the other two geodesic boundary components of the pair of pants P. Let η_i_13^X (resp. η_i_13^Y)  be the unique simple geodesic arc connecting γ_i_1 and γ_i_3 perpendicularly on X (resp. Y). Using (<ref>) and (<ref>), we see that

    cosh^2ℓ_X(η_i^X) 
       =   sinh^2 ℓ_X(γ_i_3)/2sinh^2 ℓ_X(η_i_13^X)
       =   sinh^2 ℓ_X(γ_i_3)/2[(coshℓ_X(γ_i_3')/2+coshℓ_X(γ_i_1)/2coshℓ_X(γ_i_3)/2)^2/sinh^2ℓ_X(γ_i_1)/2sinh ^2ℓ_X(γ_i_3)/2-1]
       =   sinh^-2ℓ_X(γ_i_1)/2(cosh^2 ℓ_X(γ_i_3')/2+cosh^2 ℓ_X(γ_i_1)/2+cosh^2 ℓ_X(γ_i_3)/2. 
              . + 2coshℓ_X(γ_i_3')/2coshℓ_X(γ_i_1)/2coshℓ_X(γ_i_3)/2-1).

 Similarly, we have

    cosh^2ℓ_Y(η_i^Y)  
       =   sinh^-2ℓ_Y(γ_i_1)/2(cosh^2 ℓ_Y(γ_i_3')/2+cosh^2 ℓ_Y(γ_i_1)/2+cosh^2 ℓ_Y(γ_i_3)/2. 
              . + 2coshℓ_Y(γ_i_3')/2coshℓ_Y(γ_i_1)/2coshℓ_Y(γ_i_3)/2-1).

Since ℓ_X(γ_i)≤ℓ_Y(γ_i),  this implies that

    coshℓ_X(η_i^X)/coshℓ_Y(η_i^Y)≤sinhℓ_Y(γ_i_1)/2/sinhℓ_X(γ_i_1)/2.


Subcase 2-1: ℓ_X(γ_i_1)≥<ref>.  In particular, ℓ_X(γ_i_1)=ℓ_Y(γ_i_1). Hence by (<ref>), we see that ℓ_X(η_i^X)≤ℓ_Y(η_i^Y). Combining with  (<ref>), and the assumption that γ_i_1=γ_i_2, we infer that ℓ_X(ξ_i^X)≤ℓ_Y(ξ_i^Y).

Subcase 2-2: ℓ_X(γ_i_1)< <ref>.  Then ℓ_Y(γ_i_1)= <ref>.  It follows from (<ref>) and (<ref>) that

    ℓ_X(η_i^X)≥log2/sinh<ref>/2   and   ℓ_Y(η_i^Y)≥log2/sinh<ref>/2.

Furthermore,  from  (<ref>), we see that

    coshℓ_X(η_i^X)/coshℓ_Y(η_i^Y)≤sinh<ref>/2/sinh(X)/2.

Applying an argument similarly as in the Subcase 1-2 yields 

    ℓ_X(ξ^X_i)/ℓ_Y(ξ^Y_i)≤<ref>· |log(X)|

for some constant universal constant . 

In summary, the first item of the claim follows from the discussion in Subcase 1-1 and Subcase 2-1. The second item of the claim follows from the discussion in Subcase 1-2 and Subcase 2-2 by setting <ref>=max{<ref><ref>/<ref>,<ref>}. The proof of the claim, and hence of the lemma, is now complete. 


	Instead of using strip deformations, we could also use the second item of Lemma <ref> or <cit.> to prove Lemma <ref>. The reason for using strip deformation is that strip deformation is intuitive and straightforward.
	





		


§ EXISTENCE OF MINIMUM IN MODULI SPACE

The goal of this section is to prove Theorem <ref>, which states that the algebraic intersection form K(X) has a minimum in the moduli space. 


We start the proof with the following lemma.


			Let X∈_g be a hyperbolic surface with a simple closed geodesics γ of length at most  <ref>. If α is an  oriented simple closed geodesic with |(γ,α)|<ı(γ,α), then there exist oriented simple  closed geodesics α_1,⋯,α_k such that 
			
				
  * α_1∪α_2∪⋯∪α_k is homologous to  α, and
				
  * ℓ_X(α_1)+ℓ_X(α_2)+⋯+ℓ_X(α_k)≤ℓ_X(α), and
				
  * |(γ,α_i)|=ı(γ,α_i) for every α_i.
			
		
		
		  Notice that γ cuts α into  ı(γ,α) segments.  Let α_1 be a component of α\γ such that the algebraic intersection from γ (for any choice of orientation) to α  has different signs at the endpoints of α_1.  Notice that α_1 cuts γ into two subsegments. Choose one of them with appropriate orientation, say γ_1, such that α_1∪γ_1 is an oriented simple closed curve.  Let α_1 be a  simple closed geodesic homotopic to α_1∪γ_1.  Let γ_1^- be the segment underlying γ_1 with opposite orientation. Let α_1' be the oriented closed geodesic homotopic to (α\α_1) ∪γ_1^-. From the construction, we see that 
	
		
  * α_1∪α_1'  is homologous to α, 
		
  * ı(α_1∪α_1',γ)≤ı(α,γ)-2.
	
	   Moreover, the assumption that ℓ(γ)≤<ref> implies that 
	
    ℓ_X(α_1) +ℓ_X(α_1') ≤ℓ_X(α).

	To see this, 	let ξ be  the component (α_1∪γ_1)∩(γ) which contains γ_1. Then ξ is homotopic to a subsegment ξ_0 of the boundary component of (γ). The assumption ℓ(γ)≤<ref> implies that γ admits a collar neighbourhood (γ) of width 
	
    arcsinh1/sinh (ℓ_X(γ)/2)≥√(2).

	Furthermore, each of the boundary components of (γ) has length at most 2√(2) (see Section <ref>).  
	Then the length Length_X(ξ\γ_1) of ξ\γ_1, which consists of two geodesic arcs connecting the boundary of (γ) to γ,  and the length Length_X(ξ_0) of ξ_0 satisfy:
	
    Length_X(ξ\γ_1)≥ 2√(2)≥Length_X(ξ_0).

	Hence, we could homotopy α_1∪γ_1 to a new curve by replacing ξ with ξ_0  so that the resulting curve, and hence its geodesic representative α_1, is of length less than ℓ_X(α_1). Similarly, we have ℓ_X(α_1')≤ℓ_X(α\α_1), completing the proof of (<ref>).
	
		
	
	
	Note that α_1 is  simple while α_1' is possibly not. If α_1' is not simple, then we apply a smoothing operation at each self-intersection point of α_1' (see Figure <ref>). This gives us several oriented simple closed geodesics α_11',⋯,α_1k' such that
	
		
  * α_11'∪⋯∪α_1k'  is homologous to α_1',
		
  * ∑_j ı(γ,α_1j')≤ı(γ, α_1'),
		
  * ∑_j ℓ_X(α_1j')≤ℓ_X(α_1').
	
	
	Set A_1:={α_1,α_11',⋯,α_1k'}. Then, 
		
		
  * ∪_α'∈ A_1α' is homologous to α,
		
  * ∑_α'∈ A_1ı(α',∪γ_i)≤ı(α,γ)-2,
		
  * ∑_α'∈ A_1ℓ_X(α')≤ℓ_X(α).
	
	 Inductively, suppose that A_i-1 has been constructed. We apply the above construction for each of α'∈ A_i-1 with |(α',γ)|<ı(α',γ). This yields a set A_i of oriented simple closed geodesics such that 
	
		
  * ∪_α'∈ A_iα' is homologous to α,
		
  * ∑_α'∈ A_iı(α',∪γ_i)≤ı(α,γ)-2i,
		
  * ∑_α'∈ A_iℓ_X(α')≤ℓ_X(α).
	
	Since for each i, ∑_α'∈ A_iı(α',∪γ_i)≥ |(α,γ)|,  we see that the construction stops after m≤ı(α,γ)-|(α,γ)|/2 steps. The resulting set A_m satisfies the required properties. 
		

Note that the defining equation  (<ref>) of the algebraic intersection form K(· ) is also well-defined for boarded surfaces. This allows us to extend the definition of K(·) to hyperbolic surfaces with geodesic boundary components. 


	Let X∈ T_g be a hyperbolic surface with a collection of disjoint separating simple closed geodesics  γ_1,⋯,γ_k of length at most <ref>.  Then,	
    K(X)=max_i K(X_i)

	where X_1, ⋯, X_k+1 are  components of X\(γ_1∪⋯∪γ_k).


	Let X_1,X_2,⋯, X_k+1 be the components of X\(γ_1∪⋯∪γ_k).  Let α and β be two simple closed geodesics. Since γ_i is a separating simple closed geodesic, it follows that (α,γ_i)=(β,γ_i)=0. Applying  Lemma <ref> to each γ_i and α,	we see that there exist 2k+2  sets Ω_1,⋯,Ω_k+1, Ψ_1,Ψ_1,⋯,Ψ_k+1 of oriented simple closed geodesics  such that 
		
		
  * ∪_α'∈Ω_iα'⊂ X_i and ∪_β'∈Ψβ'⊂ X_i for each i
		
  * ∪_i∪_α'∈Ω_iα'   is homologous to α and ∪_i∪_β'∈Ψ_iβ'   is homologous to  β,
		
  * ∑_i∑_α'∈Ω_iℓ_X(α')≤ℓ_X(α) and ∑_i∑_β'∈Ψ_iℓ_X(β')≤ℓ_X(β).
	
	Then,
	
    |(α,β)|/ℓ_X(α)ℓ_X(β)   ≤   |∑_i(∪_α'∈Ω_iα',∪_β'∈Ψ_iβ')|/(∑_i∑_α'∈Ω_iℓ_X(α'))(∑_i∑_β'∈Ψiℓ_X(β'))
       ≤   max_i|(∪_α'∈Ω_iα',∪_β'∈Ψ_iβ')|/(∑_α'∈Ω_iℓ_X(α'))(∑_β'∈Ψiℓ_X(β'))
       ≤   max_imax_α'∈Ω_i,β'∈Ψ_i|(α',β')|/ℓ_X(α')ℓ_X(β')
       ≤   max_i K(X_i).
	
		Consequently, 
	
    K(X)=sup_α,β|(α,β)|/ℓ_X(α)ℓ_X(β)≤max_i K(X_i).

	
	
	
		
		


 We also need the following estimate, which is essentially the lower bound of <cit.>. For reader's convenience, we provide a proof.


Let X∈_g be a hyperbolic surface with _h(X)≤<ref>. Then,
	
    K(X)≥1/<ref>·  g·_h(X)(g+|log_h(X)|)

	for some universal positive constant . 


	Let γ be one of the shortest nonseparating simple closed geodesics. In particular, ℓ_X(γ)=_h(X). By <cit.>, there exists a pants decomposition Γ which contains γ such that
	
    ℓ_X(γ')≤ 6√(3π)(g-1)

	for every γ'∈Γ. Let β be a simple closed geodesic that intersects γ exactly once and intersects every other simple closed curve in Γ at most once. Next, we shall estimate the length of β from above.
	Let P in X\Γ be an arbitrary pair of pants with boundary components γ_i,γ_j,γ_k. Let δ be  a geodesic arc  in P perpendicular to γ_i and γ_j with γ_i≠γ_j. Then by (<ref>), we see that 
	
    coshℓ_X(δ)   =   cosh(ℓ_X(γ_k)/2)+cosh(ℓ_X(γ_i)/2)cosh(ℓ_X(γ_j)/2/sinh(ℓ_X(γ_i)/2)sinh(ℓ_X(γ_j)/2
       ≤   2cosh^2(3√(3π)(g-1))/sinh^2(_h(X)/2).

	Since _h(X)≤<ref>, it follows that there exists a positive constant  such that sinh(ℓ_X(α)/2)≥<ref>ℓ_X(α).
	Hence
	
    ℓ_X(δ)≤log2+logcoshℓ_X(δ)≤ 4log 2+6√(3π)(g-1)+2log1/<ref>+2log1/_h(X).

	Since β intersects each of {γ_i} at most once, we may homotopic β to a new curve 
	
    δ_1*ξ_1*δ_2*ξ_2*⋯ *δ_m*ξ_m
	
	with m≤ |Γ|=3g-3,
	where δ_i is a geodesic arc perpendicularly connecting distinct boundary components of some pair of pants in X\Γ and ξ_i is a subarc of some of {γ_i}.  Combining with the estimate above about   the length of δ_i, we see that
	
    ℓ_X(β)≤∑_i (ℓ_X(δ_i)+ℓ_X(ξ_i))≤<ref>·  g (g+log1/_h(X))
 
	for some universal constant .
	Combined with the definition of the algebraic intersection form, this implies that
	
    K(X)   ≥   |(γ,β)|/ℓ_X(γ)ℓ_X(β)
       ≥   1/<ref>·  g·_h(X)(g+|log_h(X)|).

	This completes the proof.


As a direct consequence, we have

	Given a hyperbolic surface X∈_g with homologically systolic length _h(X)≤(log g/g)^2,  then 
K(X)≥<ref>/(log g)^2 for some universal positive constant .



	Let X_n∈_g be a sequence of hyperbolic surfaces realizing the infimum of the algebraic intersection form, that is 
	
    lim_n→∞ K(X_n)=inf _X∈_g K(X).

	If {X_n}_n is contained in some compact subset of _g, then the theorem follows by taking any accumulation point of {X_n}.  We now assume that X_n→∞ as n→∞.  By Lemma <ref>, we see that the length of nonseparating simple closed geodesics are bounded from below, say at least >0. That X_n goes to infinity implies that the length of some separating simple closed geodesic goes to zero. 	Let Γ_n be the set of separating simple closed geodesics on X_n of length strictly less <ref>.  By Lemma <ref>, there exists a hyperbolic surface X_n' such that
	
		
  * ℓ_X_n'(γ)=<ref> for every γ∈Γ_n.
		
  * ℓ_X_n'(α)≥ℓ_X_n(α)≥<ref> for every simple closed geodesic  α disjoint from any γ∈Γ_n.
	
	Every simple closed geodesic on X'_n that intersects some curve γ∈Γ_n is of length at least twice the width of the collar neighborhood (γ) of γ.
	Therefore, the sequence {X_n'} is contained in a compact subset of _g.   Furthermore, by Lemma <ref>, it follows that  K(X_n')≤ K(X_n). Hence,  by (<ref>), the limit of any convergent subsequences of {X_n'} attains the minimum min_X∈_gK(X). This completes the proof.




  


§ ESTIMATE OF MINIMAL INTERSECTION FORM


In this section, we shall estimate the minimum of algebraic intersection form in the moduli space, proving Theorem <ref>.  Before that, let us recall the following result by Balacheff, Parlier, and Sabourau.


	Let M be a Riemannian surface of genus g and area 4π(g-1). Then there exist 2g closed curves which induce a basis in H_1(M,) such that
	
    ℓ_M(α_k)≤2^16/min{_h(M),1}·log (2g-k+2)/2g-k+1· g,

	where _h(M) represents the homologically systolic length of M.

As a direct consequence, we have

	Let M be a Riemannian surface of genus g and area 4π(g-1). Then 
	
    K(M)≥2^-34(min{_h(M),1}/log g)^2,

		where _h(M) represents the homologically systolic length of M.


	Let α_1,⋯,α_2g be the curves obtained from Theorem <ref>. Since they induce a basis in H_1(X,), there exist two curves  α_i_1 and α_i_2 with 1≤ i_1≤ i_2≤ g+1 such that |(α_i_1,α_i_2)|=1.  Hence,
	
    K(X)   ≥   |(α_i_1,α_i_2)|/ℓ_X(α_i_1)ℓ_X(α_i_2)≥1/ℓ_X(α_g+1)ℓ_X(α_g+1)
       ≥   (min{_h(X),1}/2^16)^2 1/log ^2(g+1)
       ≥   (min{_h(X),1}/2^16)^2 1/4log ^2 g=1/2^34(min{_h(X),1}/log g)^2 .









	Upper bound. For any X∈_g,  let (X) (resp. _h(X)) be the length of any (resp. homological) systole of X.  By <cit.>, we have
	
    K(X)≤9/_h(X)^2≤9/(X)^2.

 Hence,
	
    min_X∈_g K(X)≤9/max_X∈_g(X)^2.

	Recall that max_X∈_g(X) ≥ Clog g for some universal constant C>0 (<cit.>, see also <cit.>; see <cit.> for a summary). Consequently, we have
	
    min_X∈_g K(X)≤C/(log g)^2.

	 for some universal constant C>0. 
	
	
	Lower bound. Let X∈_g be hyperbolic surface realizing the minimum: 
	
    K(X)=min_Y∈_gK(Y).
	
	 If _h(X)≥<ref> or _h(X)<(log g/g)^2, then by Proposition <ref> or Corollary <ref>, we see that K(X)≥ C (log g)^-2 for some universal constant C.  In the following, we assume that 
	 
    (log g/g)^2≤_h(X)≤<ref>.

	 	By Lemma <ref> and Lemma <ref>, we may assume further that every separating simple closed geodesic on X has length at least <ref>. In particular, 
	 		 
    (log g/g)^2< (X)< <ref>.
 	
	 	Let {γ_1,⋯,γ_k} be the set of simple closed geodesics on X whose length are strictly less than <ref>:
	 	
    ℓ_X(γ_i)<<ref>.

	 	In particular,  each γ_i is nonseparating.
	 
	
   
   
		Let Y be the surface obtained in Lemma <ref>. Then _h(Y)≥(Y)=<ref>.  Combined with Proposition <ref>, this implies that
		
    K(Y)≥2^-34<ref>^2/
    			(log g)^2.

	By Theorem <ref>,  there exist 2g oriented closed geodesics α_1,,⋯,α_2g on Y which induces a basis of H_1(Y,) such that 
	
    ℓ_Y(α_j)≤2^17/<ref>·log g,    for 1≤ j≤ g+1.

	  Being a basis of H_1(Y,) implies that two of the first g+1 curves, say α_1 and α_2, satisfy |(α_1,α_2)|= 1.  Considering the intersection between α_i and γ, there are two cases.
      
      Case 1:  ∑_j(|(α_1,γ_j)|+|(α_2,γ_j)|)>0.  In this case, there exists one pair (α_i,γ_j) with positive algebraic intersection,  say (α_1,γ_1).  Then by Lemma <ref>, it follows that
      
    K(X)   ≥   |(α_1,γ_1)|/ℓ_X(α_1)ℓ_X(γ_1)
       ≥   1/<ref>|log(X)|ℓ_Y(α_1)ℓ_X(γ_i)    ( by Lemma <ref> )
       ≥   1/<ref>2^18(log g)^2    ( by (<ref>), (<ref>), and (<ref>)).



    Case 2:  ∑_j(|(α_1,γ_j)|+|(α_2,γ_j)|)=0.
	For each α_i, applying Lemma <ref> for the each of the pair (α_i,γ_1) yields a set A_i^1 of oriented simple closed geodesics such that 
	
		
  * ∪_α'∈ A_i^1α' is homologous to  α_i,
		
  * ∑_α'∈ A_i^1ℓ_Y(α')≤ℓ_Y(α_i),
		
  * ∪_α'∈ A_i^1α' is disjoint from γ_1.
	
	Now inductively, suppose that for 1≤ j≤ k-1, we have constructed the set A_i^j satisfying the following:
	
		
  * ∪_α'∈ A_i^jα' is homologous to  α_i,
		
  * ∑_α'∈ A_i^jℓ_Y(α')≤ℓ_Y(α_i),
		
  * ∪_α'∈ A_i^jα' is disjoint from any of γ_1,⋯,γ_j.
	
	
	
	 For each α'∈ A_i^j, we  apply Lemma <ref> for  the pair (α',γ_j+1), yielding a set A_i^j+1 of oriented simple closed geodesics such that
		
		
  * ∪_α”∈ A_i^j+1α”  is homologous to ∪_α'∈ A_i^jα', which is in turn homologous to   α_i by the inductive assumption,
		
  * ∑_α”∈ A_i^j+1ℓ_Y(α”)≤∑_α'∈ A_i^jℓ_Y(α'), where the latter is at most ℓ_Y(α_i) by the inductive assumption,
		
  * ∪_α'∈ A_i^j+1α' is disjoint from any of γ_1,⋯γ_j,γ_j+1.
	
After k steps, we arrive at two sets A_1^k and A_2^k of oriented simple closed geodesics such that for each i, 
	
		
  * ∪_α'∈ A_iα' is homologous to α_i,
		
  * ∑_α'∈ A_iℓ_Y(α')≤ℓ_Y(α_i), and
		
  * ∪_α'∈ A_iα' is disjoint from any of  γ_1,⋯,γ_k.
	
	Combining with the fact that |(α_1,α_2)|=1, we infer that there exists α_i'∈ A_i^k such that
	
		
  * |(α_1',α_2')|>0,
		
  * ℓ_Y(α_1')≤ℓ_Y(α_1) and ℓ_Y(α_2')≤ℓ_Y(α_2),
		
  * α_1'∪α_2' is disjoint from any of  γ_1,⋯,γ_k.
	
	Combined with Lemma <ref> and (<ref>), this implies that
	
    ℓ_X(α_i')≤ℓ_Y(α_i')≤2^17/<ref>log g.

Hence 

    K(X)≥|(α_1',α_2')|/ℓ_X(α_1')ℓ_X(α_2')≥2^-34<ref>^2/(log g)^2.

This completes the proof.
	



§ ASYMPTOTIC BEHAVIOR IN MODULI SPACE

In this section, we will describe the asymptotic behavior of the algebraic intersection form  K(X) as the homologically systolic length  _h(X) goes to zero.  The key is to find good representatives of oriented simple closed curves approximating K(X).


 §.§ Finding good representatives of curves

We start with the following lemma.

			Let X∈_g be a hyperbolic surface with γ_1,⋯,γ_k being the set of nonseparating simple closed geodesics of length at most <ref>. For any oriented simple closed geodesic α on X, there exists a set A of oriented simple  closed geodesics  such that 
			
				
  * ∪_α'∈ A α' is homologous to  α, and
				
  * ı(α',γ_j)∈{0,1} for every α'∈ A and every γ_j,
				
  * ∑_α'∈ Aℓ_X(α') ≤ℓ_X(α) (1+max_iℓ_X(γ_i)/|logℓ_X(γ_i) |).
			
		
		
		
			Applying Lemma <ref> for every γ_i, we see that there exist oriented simple  closed geodesics α_1,⋯,α_m such that 
			
				
  * α_1∪α_2∪⋯∪α_m is homologous to  α, and
				
  * ℓ_X(α_1)+ℓ_X(α_2)+⋯+ℓ_X(α_m)≤ℓ_X(α), and
				
  * |(γ_j,α_i)|=ı(γ,α_i) for every α_i and every γ_j.
			
			In particular, the third property above means that for each pair (i,j), α_i and γ_j have the same algebraic intersection sign at all of their intersection points. Next, for each α_i, we shall construct oriented simple closed geodesics with required properties. 
			
			
			Let us start with α_1.   Suppose that ı(α_1,γ_1)≥2. Otherwise, we move on to consider α_2.  Let α_1:[0,ℓ_X(α_1)]→ X be a length parametrization of α_1. Choose an orientation of γ_1 such that α_1 enters the  collar neighbourhood (γ_1) of γ_1 from the left side (see Section <ref> for the definition of (γ_1)).  This is possible due to the property (iii) above. Furthermore, we assume that α_1(0) is on the left boundary component of the collar neighbourhood of γ_1, with repect to the orientation of γ_1.  Set t_1=0. Let s_1∈ (0,ℓ_X(α_1)) be the first moment that α_1 leaves   the  collar neighbourhood of γ_1. Let t_2 ∈ (s_1,ℓ_X(α_1)) be first moment that α_1 enters the  collar neighbourhood of γ_1 again. Let s_2∈ (t_2,ℓ_X(α_1)) be the first moment that α_1 leaves the  collar neighborhood of γ_1 after t_2.  Let c_i∈(t_i,s_i) be the moment that α_1|_[t_i,s_i] intersects γ_1.  Let γ_1:[0,ℓ_X(γ_1)]→ X be a length parametrization of γ_1. For s,t∈[0,ℓ_X(γ)), let γ_1|_[s,t] be the oriented segment from γ_1(s) to γ_1(t) with respect to the orientation of γ_1.  Let γ_1|_[s,t]	 be the underlying segment of γ_1|_[s,t]	  with the opposite orientation.	
			 Let d_i∈ [0,ℓ_X(γ_1)) be the moment that γ_1 intersects the segment α_1|_[t_i,s_i].   Let α̂_11 and α̂_12 be respectively the oriented simple closed curves defined as: 
			
    α̂_11:= α_1|_[t_1, c_1]*γ_1|_[d_1,d_2]*α_1|_[c_2,t_1]

			and
			
    α̂_12:= α_1|_[t_2, c_2]*γ_1|_[d_1,d_2]*α_1|_[c_1,t_2] .
	
		  Let α'_11 and α'_12 be respectively the oriented simple closed geodesic homotopic to α̂_11 and α̂_12. Then
			 
				
  * α'_11∪α'_12 is homologous to α_1,
				
  * 1≤ı(α'_11,γ_1)≤ı(α_1,γ_1)-1,  1≤ı(α'_12,γ_1)≤ı(α_1,γ_1)-1,
				
  * ℓ_X(α'_11)+ℓ_X(α'_12)≤ℓ_X(α_1)+ 2ℓ_X(γ_1),
			
			where the last inequality follows from the fact that both γ_1|_[d_1,d_2] and γ_1|_[d_1,d_2] are of length less than ℓ_X(γ_1).
			
			Repeating the above construction for each of α_11 and α_12 and the new resulting curves yields a set A_11 of oriented simple closed geodesics such that
			 
				
  * ∪_α'∈ A_11 α' is homologous to α_1,
				
  * ı(α',γ_1)=1 every α'∈ A_11,
				
  * ∑_α'∈ A_11ℓ_X(α')≤ℓ_X(α_1)+ 2ı(α_1,γ_1)ℓ_X(γ_1).
			
			
			Similarly, for every other α_i with ı(α_i,γ_1)≥ 1, there exists a  set A_1i of oriented simple closed geodesics such that
			 
				
  * ∪_α'∈ A_1i α' is homologous to α_i,
				
  * ı(α',γ_1)=1 every α'∈ A_1i,
				
  * ∑_α'∈ A_1iℓ_X(α')≤ℓ_X(α_i)+ 2ı(α_i,γ_1)ℓ_X(γ_1).
			
			Combining with the property (i), (ii), (iii) of α_1,⋯,α_m at the beginning of the proof, we arrive at a set A_1=A_11∪⋯∪ A_1m of oriented simple closed geodesics such that
			 
				
  * ∪_α'∈ A_1 α' is homologous to α,
				
  * ı(α',γ_1)∈{0,1} every α'∈ A_1,
				
  * ∑_α'∈ A_1ℓ_X(α')≤ℓ_X(α)+ 2ı(α,γ_1)ℓ_X(γ_1).
			
	Here the second item has an additional possibility ı(α',γ_1)=0 is due to the fact that some of α_1,⋯,α_m may be disjoint from γ_1.
		
			
			Applying the above construction for  γ_2, γ_3,⋯,γ_k finitely many times, we arrive at a  set A of oriented simple closed geodesics such that
			
				
  * ∪_α'∈ A α' is homologous to α,
				
  * ı(α',γ_j)∈{0,1} for every α'∈ A and every γ_j,
				
  * ∑_α'∈ Aℓ_X(α')≤ℓ_X(α)+ 2∑_jı(α,γ_j) ℓ_X(γ_j).
			
      The assumption ℓ_X(γ_j)≤<ref> implies that γ_j admits  a collar neighbourhood of width 
	
    arcsinh1/sinh (ℓ_X(γ_j)/2)≥ |logℓ_X(γ_j)|.

     Hence, the length ℓ_X(α) satifies 
       
    ℓ_X(α)≥∑_j 2·ı(α,γ_j)· |logℓ_X(γ_j)|.

      Therefore,
      
    2∑_jı(α,γ_j) ℓ_X(γ_j)   ≤   max_j ℓ_X(γ_j)/|logℓ_X(γ_j)|∑_j 2·ı(α,γ_j)· |logℓ_X(γ_j)|
       ≤   max_j ℓ_X(γ_j)/|logℓ_X(γ_j)|·ℓ_X(α).

      Accordingly, 
   
    ∑_α'∈ Aℓ_X(α')≤(1+  max_j ℓ_X(γ_j)/|logℓ_X(γ_j)|)·ℓ_X(α).

   This completes the proof.
		
		
		
		Next, we shall compare curves that wrap many times about short curves with those that wrap a few times. 	As before, let  γ_1,⋯,γ_k be the set of nonseparating simple closed geodesics of length at most   <ref> on X∈_g.  Recall that the collar neighbourhood  (γ_i) of every short curve γ_i is foliated by its perpendiculars, geodesic arcs connecting one boundary component to the other and intersects γ_i perpendicularly (see Section <ref>). 		Let α_0 be an oriented simple closed geodesic on X which crosses γ_i at most once and which crosses every perpendicular of (γ_i) at most once.    Let (m_i)_1≤ i≤ k be a k-tuple of integers such that m_i≠0 if and only if ı(α_0,γ_i)=1.  Let α be a composition of m_j-th Dehn twist T^m_j_γ_j of α_0 around  γ_j: 
		
    α=T^m_k_γ_k∘⋯∘ T^m_1_γ_1(α_0).
 
		
		Let X, γ_1,⋯,γ_k, α_0, and α be as above.  Then, after choosing appropriate orientation for each γ_i, we have
			
				
  * α_0∪ m_1γ_1∪⋯∪ m_kγ_k is homologous to  α, and
				
  * ℓ_X(α_0)+|m_1|ℓ_X(γ_1)+⋯+|m_k|ℓ_X(γ_k)≤ℓ_X(α) (1+max_i <ref>/|logℓ_X(γ_i)|),
				for some universal constant .
		
		
		
		The first item is straightforward. For the second item, consider the collar neighborhood (γ_i) of γ_i. 
		Note that by assumption, α crosses γ_i at most once. Each of α_0∩(γ_i) and α∩(γ_i)  is either an empty set or a single segment. Moreover, the segment (if any) α_0∩(γ_i) crosses every perpendicular of (γ_i) at most once, and hence α∩(γ_i) crosses every perpendicular of (γ_i) at most |m_i|+1 times. 
		Then,  by (<ref>), we see that 
		
    |	ℓ_X(α∩(γ_i) )- (ℓ_X(α_0∩(γ_i)+ |m_i|ℓ_X(γ_i))|≤ C

		for some universal constant C. Let X_:=X\((γ_1)∪⋯∪(γ_k)) be the thick part of X. Note that each of α_0∩ X_ and α∩ X_  has ∑_iı(α,γ_i) segments. Furthermore, they are homotopic to each other relative to the boundary components of ∪_i(γ_i). Since ℓ_X(γ_i)≤<ref>, it follows that each boundary component of ∪_i(γ_i) is of length most 2√(2) (see Section <ref>). Therefore, 
		
    |ℓ_X(α_0∩ X_)- ℓ_X(α∩ X_)|≤ 2√(2)∑_i ı(α,γ_i).

		On the other hand, the assumption γ_i≤<ref> implies that the width of the collar neighbourhood (γ_i) satisfies
	
    arcsinh1/sinh (ℓ_X(γ_i)/2)≥ |logℓ_X(γ_i)|.

Hence, 
		
    ℓ_X(α)≥∑_iℓ_X(α∩(γ_i))≥ 2∑_i ı(α,γ_i)|logℓ_X(γ_i)|.

		Combining the estimates above yields
		
    ℓ_X(α_0)+∑_i |m_i|ℓ_X(γ_i)   ≤   ℓ_X(α)+∑_i ı(α,γ_i)(2√(2)+C)
       ≤   ℓ_X(α) (1+max_i2√(2)+C/|logℓ_X(γ_i)|).

		This completes the proof.
		
		
		 
		 For any constant ϵ>0, let  be a constant depending only ϵ such that  
		 
    <ref><1, 1/|log<ref>|<ϵ,  <ref>/|log<ref>|<ϵ.

		   Let X∈_g be a hyperbolic surface with _h(X)<  <ref>. Let α and β be a pair of oriented simple closed geodesics on X such that 
		 
    (1-ϵ) · K(X)≤|(α,β)|/ℓ_X(α)ℓ_X(β).

		 Let Γ:={γ_1,⋯,γ_k} be the set of nonseparating simple closed geodesics on X of length at most <ref>:
		 
    ℓ_X(γ_j)≤<ref>.

			Let A (resp.  B) be the set of oriented simple closed geodesics obtained by applying Lemma <ref> to α (resp. β).  Then
			
    |(α,β)|/ℓ_X(α)ℓ_X(β)   ≤   |∑_α'∈ A, β'∈ B(α',β')|/(∑_α'∈ Aℓ_X(α'))(∑_β'∈ Aℓ_X(β'))· (1+max_jℓ_X(γ_j)/|logℓ_X(γ_i)|)^2
    			
       ≤   max_α'∈ A, β'∈ B|(α',β')|/ℓ_X(α')ℓ_X(β')· (1+ϵ) ^2 .    ( by (<ref>), (<ref>))

			Note that 
						 
    |(α',γ_i)|=ı(α',γ_i)≤1,   			 	|(β',γ_i)|=ı(β',γ_i)≤ 1.

			 For each γ_i∈Γ, let T^n_γ_i be the n-th Dehn twist around γ_i. Let α” be one of the shortest simple closed geodesics in the Dehn twists orbit
			
    {	T^n_1_γ_1∘ T^n_2_γ_2∘⋯∘ T^n_k_γ_k (α'):n_i∈ℤ}.
	
			Assume that α”=	T^n_1_γ_1∘⋯∘ T^n_2_γ_2∘ T^n_k_γ_k (α').
			Similarly define β”:=T^m_1_γ_1∘ T^m_2_γ_2∘⋯∘ T^m_k_γ_k (β'). Being shortest in their Dehn twists orbits implies that both α” and β” intersect every perpendicular of the collar neighbourhood (γ_i) at most once.  Applying  Lemma <ref> to (α”,α') and (β”,β') independently, we see that
			
    |(α',β')|/ℓ_X(α')ℓ_X(β')(1+ϵ)^-2
       ≤   |(α',β')|/ℓ_X(α')ℓ_X(β')·(1+max_j <ref>/|logℓ_X(γ_j)|)^-2    ( by (<ref>), (<ref>))
    				
       ≤   |(α”∪ (∪_i n_iγ_i)),β”∪(∪_j m_jγ_j))|/(ℓ_X(α”)+∑_i|n_i|ℓ_X(γ_i))(ℓ_X(β”)+∑_j|m_j|ℓ_X(γ_j))
       ≤   ∑_j |m_j| · |(α”,γ_j)|+∑_i|m_i|· |(β”,γ_i))|+|(α”,β”)|/∑_j |m_j|ℓ_X(α”)ℓ_X(γ_j) +∑_i |n_i|ℓ_X(β”)ℓ_X(γ_i)+ℓ_X(α”)ℓ_X(β”)
       ≤   max_i,j{|(α”,γ_j)|/ℓ_X(α”,γ_j),|(β”,γ_i)|/ℓ_X(β”)ℓ_X(γ_i), |(α”,β”)|/ℓ_X(α”)ℓ_X(β”)}
       ≤   max{max_i1/ℓ_X(α_γ_i)ℓ_X(γ_i), |(α”,β”)|/ℓ_X(α”)ℓ_X(β”)},
	
			where the last inequality follows from (<ref>) and   the assumption that α_γ_i is a shortest nonseparating simple closed geodesic which intersects γ_i exactly once.  
			
			
			Next, we shall compare 
			
    max_i 1/ℓ_X(α_γ_i)ℓ_X(γ_i)   and   |(α”,β”)|/ℓ_X(α”)ℓ_X(β”).

			To begin, suppose γ_1 is one of the shortest one among γ_1,⋯,γ_k, that is, ℓ_X(γ_1)=_h(X) . Since α_γ_1 is one of the shortest simple closed geodesic intersecting γ_1 exactly once,  then by (<ref>), we see that
				
    ℓ_X(α_γ_1)≤<ref>·  g (g+|log_h(X)|)
	
	for some universal constant <ref>.
	Hence 	
		
    max_i	1/ℓ_X(α_γ_i)ℓ_X(γ_i)≥1/ℓ_X(α_γ_1)ℓ_X(γ_1)≥1/<ref>·  g ·_h(X)(g+|log_h(X)|)

			which goes to infinity as _h(X)→0. On the other hand,  we
			claim that 
			|(α”,β”)|/ℓ_X(α”)ℓ_X(β”)  is bounded from above.
		 To see this, let (γ_j) be the collar neighbourhood of γ_j. Let X_:=∪_i (γ_i) and X_=X\ X_  
		 be respectively the thin part and the thick part of X respectively. Then any simple closed curve in X_ is of length bounded from below by some positive constant .  Applying an argument similar as the proof of <cit.> yields that
			
    |(α”∩ X_,β”∩ X_)|/ℓ_X(α”∩ X_)ℓ_X(β”∩ X_)≤9/<ref>^2.

			Note that   ı(α”,γ_j)=ı(α',γ_j)≤ 1 and ı(β”,γ_j)=ı(β',γ_j)≤1. Furthermore, both α” and β” intersect each perpendicular of the collar neighbourhood (γ_j) at most once. Therefore, α”∩(γ_j) intersects β”∩(γ_j) at most twice:
			
    ı(α”∩(γ_j),β”∩(γ_j))≤ 2·ı(α”,γ_j)ı(β”,γ_j).

			Note that the collar neighbourhood (γ_i) of γ_i is of width at least |logℓ_X(γ_i)|. Hence,
			
    ℓ_X(α”∩ X_)≥∑_j 2ı(α”,γ_j)|logℓ_X(γ_j)|

			and 
						
    ℓ_X(β”∩ X_)≥∑_j 2ı(β”,γ_j)|logℓ_X(γ_j)|.


			 Accordingly, 
			
    |(α”∩ X_,β”∩ X_)|/ℓ_X(α”∩ X_)ℓ_X(β”∩ X_)
       ≤   ∑_j 2 ı(α”,γ_j)ı(β”,γ_j)/ℓ_X(α”∩ X_)ℓ_X(β”∩ X_)
       ≤   ∑_j 2ı(α”,γ_j)ı(β”,γ_j)/∑_j 2ı(α”,γ_j)|logℓ_X(γ_j)|·∑_j 2ı(β”,γ_j)|logℓ_X(γ_j)|
       ≤   max_j 2 / 2|logℓ_X(γ_j)|· 2|logℓ_X(γ_j)|≤1/2|log<ref>|^2.

			Therefore,
			
    |(α”,β”)|/ℓ_X(α”)ℓ_X(β”)
       ≤   |(α”∩ X_,β”∩ X_)|+|(α”∩ X_,β”∩ X_)|/(ℓ_X(α”∩ X_)+ℓ_X(α”∩ X_ ))(ℓ_X(β”∩ X_)+ℓ_X(β”∩ X_ )
       ≤   max{|(α”∩ X_,β”∩ X_)|/ℓ_X(α”∩ X_)ℓ_X(β”∩ X_), 	|(α”∩ X_,β”∩ X_)|/ℓ_X(α”∩ X_)ℓ_X(β”∩ X_)}
       ≤   max{9/<ref>^2,1/2|log<ref>|^2}.

			This completes the proof of the claim. Consequently, there exists a constant <<ref> such that if _h(X)≤<ref> then
			
    |(α',β')|/ℓ_X(α')ℓ_X(β')≤max_i {1/ℓ_X(α_γ_i)ℓ_X(γ_i)}·(1+ϵ)^2.
		
		Combining this with (<ref>) and (<ref>), we infer that
		
    K(X)≤max_i{1/ℓ_X(α_γ_i)ℓ_X(γ_i)}·(1+ϵ)^4/1-ϵ≤max_γ∈Γ{1/ℓ_X(α_γ)ℓ_X(γ)}·(1+ϵ)^4/1-ϵ,
	
		where, by assumption, Γ is the set of simple closed geodesics of length at most 1 and α_γ is one of the shortest simple closed geodesics intersecting γ∈Γ exactly once.  
			On the other hand, the definition of K(X) implies that 
			
    K(X)≥max_γ∈Γ{1/ℓ_X(α_γ)ℓ_X(γ)}.

			Hence, as _h(X)→0, we have 
			
    1≤K(X)/max_γ∈Γ{1/ℓ_X(α_γ)ℓ_X(γ)}≤(1+ϵ)^4/1-ϵ.

			The theorem then follows from the arbitrariness of ϵ. 
				
		
		
	




 §.§ Examples
   



Let γ_1,⋯,γ_3g-3 be a collection of nonseparating simple closed geodesics as in Figure <ref>. Let 0≤δ≤1 be a fixed constant. Let X_n∈_g be a hyperbolic surface such that ℓ_X_n(γ_1)=1/n^δ and ℓ_X_n(γ_i)=1/n for i≥ 2.  As n→∞, the geodesics  γ_2,⋯,γ_3g-3 become the homological systoles of X_n. The geodesic γ_1 has a collar neighbourhood of width δlog n while γ_i has a collar neighbourhood of width log n for i≥2. Any nonseparating simple closed geodesic intersecting distinct from any of {γ_i} has to intersect at least two distinct curves of {γ_i}, and hence is of length at least 2(1+δ)log n=2(1+δ)|log_h(X_n)|. By  Theorem <ref>, we see that 
		
    lim_n→∞K(X_n)·2(1+δ) ·_h(X_n) |log_h(X_n)|≤ 1.

		On the other hand, let α be one of the shortest simple closed geodesics which intersect only γ_1 and γ_2 as in Figure <ref>. Then  
		ı(α,γ_2)=1 and  |ℓ_X_n(α)-2(1+δ)log n|≤ C for some universal constant C (by (<ref>)). Therefore,
		
    K(X_n)   ≥   |(α,γ_2)|/ℓ_X_n(α)ℓ_X_n(γ_2)
       ≥   1/(2(1+δ) log n+C)·ℓ_X_n(γ_2)
       =    1/(2(1+δ) |log_h(X_n)|+C)·_h(X_n),

		where in the last equation we use the assumption that ℓ_X_n(γ_2)=_h(X_n)=1/n.
		Letting n→∞, we see that 
		
    lim_n→∞K(X_n)·2(1+δ) ·_h(X_n) |log_h(X_n)|≥  1.

		In summary, we have
		
    lim_n→∞K(X_n)·2(1+δ) ·_h(X_n) |log_h(X_n)|= 1.

		This example shows that the asymptotic behavior of the algebraic intersection form cannot be described by a single function of the homologically systolic length.



We continue with the notation in the previous example. Let γ_2' be the simple closed geodesic as in Figure <ref>, let ℓ_X_n(γ_1)=1/n^1-1/n, ℓ_X_n(γ_2')=1/n, ℓ_X_n(γ_i)=1/n for i≥ 3. Then, as n→∞, 
	
    1/ℓ_X_n(α)ℓ_X_n(γ_1)∼n^1-1/n/2 (1-1/n) |log n| ∼n/2|log n|.

On the other hand, for each homological systole γ_i (i≥ 3), any nonseparating simple closed geodesic α_γ_i intersecting γ_i  once has to intersect at least  two distinct curves among γ_2',γ_3,⋯,γ_3g-3.  Hence ℓ_X(α_γ_j)≥ 4log n. Accordingly, 

    max_3≤ i≤ 3g-31/ℓ_X_n(α_γ_i)ℓ_X_n(γ_i)≤n/4|log n|.

Therefore, by Theorem <ref>, as n→∞, 
	
    K(X)∼n/2|log n|.

This example shows that considering only homological systoles and curves intersecting them is not sufficient for describing the asymptotic behavior of the algebraic intersection form. 


 
alpha



