Mirzakhani’s recursion formula on Weil-Petersson volume and applications
We give an overview of the proof for Mirzakhani’s volume recursion for the Weil-Petersson volumes of the moduli spaces of genus hyperbolic surfaces with labeled geodesic boundary components, and her application of this recursion to Witten’s conjecture and the study of simple geodesic length spectrum growth rates.
Mirzakhani’s recursion formula on Weil-Petersson volume and applications
|Department of mathematics and statistics, University of Melbourne|
|University of Melbourne, Victoria 3010, Australia|
2000 Mathematics Subject Classification: 32G15; 14H15
Keywords: Weil-Petersson volume, Teichmüller spaces, moduli spaces, mapping class group, McShane identity, Witten’s conjecture, simple closed geodesics.
- 1 Introduction
- 2 McShane identities
- 3 Weil-Petersson volume computation
- 4 Applications
Let denote a (connected) surface with genus and punctures labeled to , so that the Euler characteristic is negative. The moduli space is a real -dimensional orbifold, whose points represent isometry classes of complete finite-area hyperbolic metrics on .
The moduli space has a natural symplectic structure given by the Weil-Petersson symplectic form , and the volume form obtained by taking the top exterior product
explicitly computing the volume of a fundamental domain for ;
identifying the Weil-Petersson form as a cohomology class, thus relating intersection numbers on with its Weil-Petersson volume.
The four-punctured sphere case was similarly derived from this first volume computation. However, it was not until Penner’s work in  that a further Weil-Petersson volume was obtained. Penner gave a fatgraph-based111Also known as ribbon-graphs to those more familiar with Kontsevich’s work. cell-decomposition of and used this to explicitly obtain that the volume of is [20, Thm. 5.2.1]. Indeed, Penner described a general strategy for computing the volumes of the moduli space for any punctured surface . In practice, however, this is intractable because:
the number of cells in grows quite quickly (by [20, Thm. B], even for the growth rate tends to ) and
the integral for the volume of each top-dimensional cell becomes difficult to exactly evaluate.
Zograf, however, explicitly expressed the Poincaré dual of the Weil-Petersson form in terms of certain divisors on the Deligne-Mumford compactification locus of , and exploited the intersection number interpretation of the Weil-Petersson volume to obtain the following recursion formula for the volume of the moduli space of -punctured spheres :
He expanded on this work in , to derive the following recursion formula for the volume of the moduli space of -punctured tori222Some readers may notice that Zograf’s volume is only half of . This is due to the fact that all once-punctured tori have hyperelliptic involutions, this is discussed in greater detail in Remark 3.2.
Then came Näätänen and Nakanishi’s work [16, 17] on computing the volumes of moduli spaces of once-punctured tori and four-punctured spheres with geodesic boundaries of length (and cone angle singularities ). Nakanishi and Näätänen found that the answer is a rational polynomial in and (or ): for the punctured torus and for the four-punctured sphere. Their computation was also based on integrating over a fundamental domain, and thus was not easily generalizable.
This was the landscape prior to Mirzakhani’s beautiful solution to the volume computation problem . In Section 1, we give the necessary background on the Weil-Petersson geometry of moduli spaces; in Section 2 we explain Mirzakhani’s proof of her McShane identities — a key element of her proof; in Section 3 we give a schematic of Mirzakhani’s proof strategy; in Section 3 we use the McShane identity to give a sample computation of the volume of ; finally, in Section 4 we give applications of Mirzakhani’s volume integration to proving Witten’s conjecture and for specifying the growth rate of simple closed geodesics on a given hyperbolic surface.
Unless otherwise specified, any surface that we consider is oriented, hyperbolic, finite area and has either geodesic or cusp borders labeled from to , where is the number of boundary components of . In terms of defining the Teichmüller space, the moduli space and intermediate moduli spaces, we only require to be a topological surface. However, it is sometimes convenient to endow with a hyperbolic structure.
1.1.1 Teichmüller space.
Let be a surface with genus and boundary components labeled to , and let be an -tuple of positive real numbers, then the Teichmüller space is:
where if and only if is isotopy equivalent to a isometry. We denote these equivalence classes by and refer to them as marked surfaces.
We adopt the convention that length means that the -th boundary component is a cusp, and we write for the Teichmüller space of a closed surfaces . Teichmüller  showed that the Teichmüller space is homeomorphic to a -dimensional open ball.
1.1.2 Mapping class groups and moduli spaces.
The group of boundary label-preserving homeomorphisms of acts on the Teichmüller space by precomposition: given ,
Note that we precompose by the inverse of so that this is a left action.
Since marked surfaces are defined up to isotopy, the normal subgroup , consisting of all homeomorphisms isotopy equivalent to the identity map , acts trivially on . Thus, we define the mapping class group
We refer to elements of as mapping classes.
Mapping class groups are finitely presentable [10, Thm. 5.3], and act discretely on the Teichmüller space. This action is almost free, in the sense that isotropy groups are finite, and the resulting quotient orbifold
is referred to as the moduli space of . Note that each point of represents a distinct isometry class of hyperbolic surfaces homeomorphic to with boundary lengths equal to . Thus, as a set, we may identify with the set
where if and only if they are isometric. We denote the isometry class of by .
1.1.3 Pairs of pants.
A pair of pants is a hyperbolic surface with genus and geodesic boundary components (allowing for cuspidal boundaries).
Theorem 1.1 ([3, Thm. 3.1.7]).
For any 3-tuple , there is a unique hyperbolic pair of pants with labeled boundaries respectively of lengths and .
Since every label-preserving self-homeomorphism on a pair of pants is homotopy equivalent to the identity map, we have:
The moduli space for a hyperbolic pairs of pants is precisely .
The only simple closed geodesics on a pair of pants are the three geodesics constituting the boundary of . Moreover, there is a unique simple geodesic arc joining boundaries and .
The three orthogeodesics , and cut up into two isometric right-angled hexagons. This tells us that there is an orientation-reversing isometry on which fixes these orthogeodesics pointwise and takes one hexagon to the other.
1.1.4 Fenchel-Nielsen coordinates.
Given a closed curve on , and a marked hyperbolic surface , there is a unique closed geodesic homotopy equivalent to . Denote this geodesic by , then the curve defines a positive valued function on Teichmüller space given by:
where is the (geodesic) length function.
A pants decomposition of a hyperbolic surface , with genus and labeled boundaries, is a maximal collection of disjoint simple closed geodesics on . As seen in Theorem 1.1, hyperbolic pairs of pants are uniquely determined by the lengths of their boundary geodesics. Thus, given a marked hyperbolic surface , the pairs of pants obtained from cutting along are uniquely determined by the lengths and the length functions on Teichmüller space.
We can recover from a pants decomposition if we know how to glue its constituent pairs of pants. For each , the endpoints of the pants seams on the pairs of pants bordered by allow us to keep track of this gluing with an element of . Moreover, since any map that fixes for is homotopy equivalent to Dehn twists of along the , we may keep track of these Dehn twists (effectively lifting up to ) and hence parameterize the entire Teichmüller space . We denote the twist parameter for by .
Given a pants decomposition of a hyperbolic surface with genus and labeled boundaries of lengths , the Fenchel-Nielsen coordinates
is a real analytic homeomorphism.
1.1.5 Weil-Petersson structure.
Fix a pants decomposition on , and consider the Weil-Petersson symplectic form on given by:
The Weil-Petersson form has a pants-decomposition independent formulation, and Wolpert showed that this -form has the above expression . This means that is invariant under the mapping class group action and descends to a symplectic 2-form on . The Weil-Petersson volume obtained by integrating the top exterior product volume form
over is finite. Wolpert  showed that smoothly extends to a closed 2-form on the Deligne-Mumford compactification of , and pairing its top exterior product with the fundamental class gives the Weil-Petersson volume. Indeed, the existence of Bers’ constant [3, Thm. 5.1.2] implies that the finite volumed set
contains a fundamental domain for the mapping class group. Although Bers’ constant is not known exactly in general, there are upper bounds for [19, Thm. 4.8] and so this does give us an explicit upper bound on the volume of .
1.2 Intermediate moduli spaces
Since the Teichmüller space is contractible, it is the (orbifold) universal cover of . Moreover, since the mapping class group is the group of deck transformations on , subgroups of the mapping class group correspond to (connected) covering spaces of the moduli space. We informally refer to these covering spaces as intermediate moduli spaces.
Now, given an ordered -tuple of simple closed geodesics on , the mapping class group acts on diagonally to produce a collection
of -tuples of simple closed geodesics on . The stabilizer
is a subgroup of the mapping class group, and corresponds to the following covering space:
where if and only if there is an isometry such that for .
1.2.1 Forgetful map.
We refer to the covering map given by
as the forgetful map, because forgets the geodesic paired with . By decorating surfaces with -tuples of geodesics, we’re able to define the following length function
1.2.2 Structure of intermediate moduli spaces.
Let denote the length of on , and let denote the connected bordered hyperbolic surfaces resulting from cutting along . Let us label/order the boundaries and denote the lengths of the borders which arise from cutting along by
and the lengths of the other borders by
Observe that and .
If consists of disjoint simple closed curves, then the multicurve length function maps surjectively onto its codomain . The preimage of consists of hyperbolic surfaces paired with an ordered -tuple of simple geodesics on . Cutting along results in hyperbolic surfaces respectively homeomorphic to . To recover from the , we only need to specify how these subsurfaces are glued together. Since the may vary over and the gluing for varies over , we obtain that:
Note that the above identification (1.1) holds for any because does not depend upon the geometry of , and we use (1.1) to describe the pullback Weil-Petersson structure on . Let denote the length of the -th geodesic in and let denote the twist parameter for . Note that is well-defined. Then, the Weil-Petersson form on is given by:
As with , we may take the top-exterior product of to obtain the Weil-Petersson volume form on :
As usual, we omit if it is the (length ) zero vector.
2 McShane identities
The following theorem is a rephrasing of McShane’s original identity for one-cusped hyperbolic tori :
For any marked one-cusped hyperbolic torus , let denote the collection of (non-peripheral) simple closed geodesics on , then
Each summand in the above series has the following geometric interpretation: the probability that a geodesic launched from the cusp in will self-intersect before hitting is precisely .
2.1 Sample application to volume integration
Fix a simple closed geodesic on and note that . Mirzakhani saw that the -pushforward of the Weil-Petersson measure for weighted by the function is precisely the Weil-Petersson volume measure induced by . To see this over a point :
This in turn means that the following integrals are equivalent:
This is our motivation for studying McShane identities: they allow us to unwrap the Weil-Petersson volume of as an integral of a function over a topologically simpler moduli space. In this particular case, cutting along results in a pair of pants with boundary lengths . Hence, (1.1) tells us that the intermediate moduli space is given by
with volume form:
Therefore, the WP-volume of should be
However, since one-cusped tori all have an order isometry called the hyperelliptic involution, we halve to derive that . This is explained more precisely in Remark 3.2.
2.2 McShane identities for bordered hyperbolic surfaces
In order to generalize this volume integration strategy, Mirzakhani generalized McShane identities for any bordered hyperbolic surface.
Theorem 2.3 (McShane identity).
Let be a bordered hyperbolic surface with genus and labeled geodesic boundaries of lengths , let
be the collection of simple closed geodesics which, along with the boundaries and , bound a pair of pants in ;
be the collection of unordered pairs of simple closed geodesics which, along with the boundary , bound a pairs of pant in .
Then, for any marked surface , we have the following identity:
where the functions and are defined by:
Note that may contain boundary geodesics.
We introduce the functions and for minor expositional reasons; they are related to Mirzakhani’s and functions by:
2.2.1 Proof strategy.
McShane identities for bordered hyperbolic surfaces may be derived by splitting the length of boundary on into a countable sum. The idea is to orthogonally shoot out geodesic rays from points on boundary and hence partition based on the behavior of these orthogeodesic rays. Specifically, starting from a point , precisely one of three things happens:
the geodesic ray hits a boundary without self-intersecting,
the geodesic ray hits or self-intersects,
this geodesic ray goes on forever without every self-intersecting.
Case 1: let denote the geodesic arc emanating from that hits the -th boundary . Then, may be fattened up to a unique (homotopy equivalent) pair of pants. The arc must be wholly contained in , or else forms a hyperbolic -gon — a geometric impossibility.
Case 2: let denote the geodesic arc emanating from up to its intersection with or its first point of self-intersection. Then, may be fattened up to a unique (homotopy equivalent) pair of pants. The arc must be wholly contained in , or else forms either a hyperbolic 2-gon or a hyperbolic triangle with internal angles strictly greater than — both geometric impossibilities.
Case 3: there are uncountably many such simple orthogeodesic rays. However, the Birman-Series geodesic sparsity theorem  tells us that they occupy a set of measure on . More accurately, the Birman-Series theorem says that the collection of simple geodesics on a closed hyperbolic surface has Hausdorff dimension . Double a bordered hyperbolic surface by gluing an isometric (but orientation-reversed) copy to along correspondingly labeled borders. Orthogeodesics rays on then glue to corresponding orthogeodesics rays on to give simple bi-infinite geodesics on the double. The Birman-Series theorem then asserts that the set of points occupied by these bi-infinite geodesics has measure on . This in turn means that the restriction of these simple bi-infinite geodesics to a collar neighborhood of boundary (as a subset of the double) occupies area. However, this collar neighborhood has the structure of an interval times , and so we see that the set of points on boundary that launch simple orthogeodesic rays has measure with respect to the length measure on the boundary.
We conclude therefore that almost every belongs either to Case 1 or Case 2, and in these two cases, the geodesic arc lies on a unique pair of pants in . This gives us a natural decomposition of the total measure of as an infinite sum over pairs of pants embedded in .
2.2.2 Orthogeodesics on pairs of pants.
Given a pair of pants with boundaries , there are precisely four simple infinite orthogeodesic rays contained in (Figure 4, second from left). The end points of these four rays partition into four intervals.
Geodesic rays launched from the interval closest to necessarily hit (before possibly self-intersecting; Figure 4, leftmost). The width of this side interval (computed with a little hyperbolic trigonometry [3, Thm. 2.3.1]) is
Terms of the form (2.10) arise in Case 1, that is: for embedded pairs of pants which contain two distinct boundary geodesics . Since pairs of pants may be given by specifying their boundary geodesics in , the collection of pairs of pants bordered by and precisely corresponds to . Replacing with and with in (2.10) produces the correct summand in Mirzakhani’s identity.
By symmetry, the above statements also hold for the interval closest to upon switching the roles of and .
Orthogeodesic rays launched from the two remaining intervals either self-intersect (before possibly leaving ; Figure 4, center-left) or hit (Figure 4, right). The width of each of these two middle intervals is:
Terms of the form (2.11) arise in Case 2, that is: for embedded pairs of pants which contain . This corresponds to . Doubling (2.11) due to there being two such intervals for produces the correct summand in the bordered McShane identity.
This completes the proof of McShane identity for bordered hyperbolic surfaces.
2.2.3 Simple infinite orthogeodesic rays.
As a minor aside, we point out that Mirzakhani also gives a detailed analysis of what occurs in Case 3.
Theorem 2.5 ([14, Thm. 4.5, 4.6]).
The set of points on corresponding to simple infinite orthogeodesics is homeomorphic to the Cantor set union countably many isolated points. Specifically, if the orthogeodesic ray emanating from
spirals to a boundary curve , then is an isolated point;
spirals to a simple closed geodesic in the interior of , then is a boundary point of the Cantor set;
does not spiral to a simple closed curve, then is a non-boundary point of the Cantor set.
Mirzakhani’s proof of the above result is slightly technical, but the result itself is geometrically unsurprising. In the course of establishing her McShane identity, we have seen that the Case 1 and Case 2 points each lie within precisely one of four (open) intervals on a certain pair of pants . Case 3 points are the points on that still remain after removing these open intervals.
When a pair of pants contains a boundary geodesics distinct from , we need to remove three of the intervals — leaving the interval closest to . The end points of the two simple infinite orthogeodesic rays wedged in between these three intervals are obviously isolated points, and these rays spiral to . All isolated points arise in this way.
If we add these isolated points to the points that we remove from , then for every pair of pants , we remove one long open interval (containing three of the original intervals) and for every pair of pants , we remove two open intervals. This process of removing intervals from is essentially akin to how the standard Cantor set is constructed, and it should be expected that the remnant collection of points is a Cantor set. Moreover, this description tells us that the boundary points of the intervals that we remove correspond to orthogeodesic rays which spiral to an interior simple closed geodesic — as was asserted in statement (2).
Note that the existence of the Case 3 Cantor set is one reason for which we needed to invoke the Birman-Series theorem. After all, the measure of a Cantor set on can take any value in .
3 Weil-Petersson volume computation
Let denote the Weil-Petersson volume of the moduli space .
3.1 Mirzakhani’s volume recursion formula
3.1.1 Derivation and .
We summarize the key steps of Mirzakhani’s volume computation procedure, while giving a step-by-step calculation of as an illustrative example.
Step 1: Rearrange the McShane identity into mapping class group orbits of ordered tuples of curves. This prepares the McShane identity in a form conducive to the integral unwrapping we saw in (2.5).
First identify elements of the form and since they both correspond to the pair of pants on bordered by . Gather the summands of the form over with the corresponding summand over to get series of the form
The remaining elements of correspond to pairs of pants whose non- borders are on the interior of . We break up the summands over into two summands of the form over and , thereby enabling us to sum over
Partition the new summation index set of ordered geodesic pairs into mapping class group orbits, and gather the summands accordingly.
To determine whether two ordered curves and are in the same mapping class group orbit, check if and respectively decompose into surfaces with topologically equivalent connected components with matching boundary labels and where matches with .
Consider a thrice-holed hyperbolic torus with boundary lengths , and recall that Mirzakhani’s McShane identity (2.3) for marked surfaces is a series summed over certain sets and . Choose arbitrary elements , then .
The ordered summation index
partitions into three mapping class orbits:
where are chosen so that excising the pairs of pants bordered by from results in a connected surface (Figure 5 center), and where are chosen so that excising the pairs of pants bordered by results in a disconnected surface (Figure 5 right). Thus, the McShane identity may be rearranged as follows:
Step 2: Integrate the McShane identity over moduli space. The left hand side is and the right hand side unwraps as integrals over various intermediate moduli spaces. Specifically, we use the fact that the pushforward of the weighted WP volume measure onto with respect to the forgetful map is precisely given by
to unwrap to . And we use the fact that the pushforward measure of the weighted WP volume measure onto with respect to the forgetful map is
to unwrap to .
Integrating (3.12) over , we obtain