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# The Eskin-Mirzakhani-Mohammadi Magic Wand

## Structure of orbits: a geometric Ratner’s theorem?

The ergodicity of the $\mathrm{SL}(2,\mathbb{R})$ action on the moduli space $\mathcal{H}$ of translation surfaces (= moduli space of Abelian differentials, under the identification we made earlier) allows us to understand generic orbits, but not of arbitrary orbits. In particular, for example, a family of flat surfaces correspond to a fixed rational polygonal billiard forms a positive-codimension subspace, about which ergodicity allows us to say nothing.

There are, however, results in dynamics / ergodic theory which classify not just almost all, but all orbits, the prototypical example being Ratner’s Theorem(s) on unipotent flows:

Theorem/s (Ratner) Let G be a connected Lie group and U be a connected subgroup generated by unipotents. Then

• for any lattice $\Gamma \subset G$ and any $x \in G / \Gamma$, the closure of the orbit $Ux \in G / \Lambda$ is an orbit of a closed algebraic subgroup of G.
• every ergodic invariant probability measure is homogeneous;
• every unipotent orbit is equidistributed in its closure.

A basic example is given by a horocycle flow on a hyperbolic manifold. These are ergodic, and so we know that almost every orbit is dense; but Ratner’s theorem tells us that in fact we have a strict dichotomy: every orbit is either closed or dense.

The hope here is for a similar result: one precise formulation of this is the following

Conjecture (“Magic Wand”) The closure of a $\mathrm{SL}(2,\mathbb{R})$-orbit of any flat surface is a complex-algebraic suborbifold. (By a theorem of Kontsevich, any $\mathrm{SL}(2,\mathbb{R})$-invariant complex suborbifold is represented by an affine subspace in cohomological period coordinates.)

## Aspects of Teichmüller theory

Recall that we have identified $\mathcal{H}$ as a space of pairs (complex structure, holomorphic 1-form). Recalling some of the plumbings of Teichmüller theory, we consider also the space of pairs (complex structure, holomorphic quadratic differential), and identify it with the cotangent bundle to the moduli space $\mathcal{M}$ of complex structures. $\mathcal{H}$ can be identified with a subspace of $\mathcal{Q}$ consisting of those quadratic differentials which can be represented as global squares of holomorphic 1-forms.

This subspace may be considered as a “unit cotangent bundle”, being invariant under the Teichmüller geodesic flow (i.e. the diagonal subgroup action induced by the $\mathrm{SL}(2,\mathbb{R})$ action on $\mathcal{H}$.)

We may check that $\mathrm{SL}(2,\mathbb{R})$ orbits in (the image of) $\mathcal{H}$ in $\mathcal{Q}$ descend to isometric maps of $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2$ to$\mathcal{M}$—i.e. the projections of these orbits are Teichmüller discs, also known as complex geodesics.

Complex geodesics may be described more directly in terms of the language of flat surfaces as follows: recall $\mathrm{SL}(2,\mathbb{R})$ orbits in $\mathcal{H}$ correspond to translation surfaces with a distinguished direction, encoded by the holomorphic 1-form; the $\mathrm{SL}(2,\mathbb{R})$ action changes the translation structure, i.e. the fundamental polygon, but not the affine structure, i.e. the resulting translation surface. Then we obtain a complex geodesic by forgetting the 1-form, i.e. forgetting the distinguished direction.

The classification of orbits is then closely related to the classification of these complex geodesics, which allows us to potentially use (even more) language and tools from Teichmüller theory.

## “Revolution in genus 2”

Kontsevich-Zorich classified strata of the moduli space using spin structures and hyperellipticity. In genus 2, the two stratum $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$ are each connected and consist entirely of hyperelliptic surfaces.

Smilie proved that closed $latex\mathrm{SL}(2,\mathbb{R})$-orbits—orbits of flat surfaces which are in some sense exceptionally symmetric—correspond exactly to orbits of Veech surfaces (see first post on polygonal billiards for a description of Veech surfaces.) The identification of closed orbits thus reduces to (or, at any rate, is equivalent to) the classification of Veech surfaces, about which some things, but not very many, are known.

McMullen proved that there is (up to ramified coverings) only one Veech surface in the stratum $\mathcal{H}(1,1)$, given by the regular decagon with identified opposite sides.

Calta and McMullen, using different methods, described all Veech surfaces in $\mathcal{H}(2)$—there is a countable family even up to ramified coverings—and gave efficient algorithms to recognize and classify these.

They also describe invariant submanifolds of intermediate dimension—intermediate between the full stratum and but larger than single closed orbits.

Finally, McMullen shows, using all of this, plus more subtle tools, that our “magic wand” conjecture is true in genus 2; the classification is in fact rather more precise, and he also obtains results about invariant measures in the spirit of Ratner’s theorems.

## Eskin-Mirzakhani-Mohammadi’s magic wand

Mirzakhani and collaborators (Eskin and Mohammadi), together with Filip, in spectacular (relatively) recent work, proved the “magic wand” conjecture, plus measure rigidity results, for all genera.

The measure rigidity result of Eskin-Mirzakhani states that any ergodic P-invariant measure (where P is a maximal parabolic subgroup, e.g. the Borel subgroup) is in fact a Lebesgue class measure on a manifold cut out by linear equations, and must be $\mathrm{SL}(2,\mathbb{R})$-invariant. This uses considerable machinery from ergodic theory: “almost 100 pages of delicate” entropy arguments, plus ideas of Benoist-Quint.

The theorem of Eskin-Mirzakhani-Mohammadi then builds on this to state that the $\mathrm{SL}(2,\mathbb{R})$-orbit closure of a translation surface is always a manifold. Moreover, the manifolds that occur are locally defined by linear equations in period coordinates, with real coefficients and zero constant term.

The proof proceeds, given the measure rigidity result, by constructing a P-invariant measure on every P-orbit closure. Here the use of P, as opposed to $\mathrm{SL}(2,\mathbb{R})$, is crucial—the former is amenable whereas the latter is not, and this allows us to use averaging methods in our construction.

(Filip’s result is needed to go from analyticity, which Eskin-Mirzakhani-Mohammadi actually gives us, to algebraicity.)

## Where can the magic wand take us?

These results allow us to say things about specific families of translation surfaces—e.g. a rational billiard table, whose orbit under the $\mathrm{SL}(2,\mathbb{R})$-action forms a high-codimension family in $\mathcal{H}$—rather than just “almost all” translation surfaces

Thus, for instance, we can prove quadratic asymptotics (exact, not just lower and upper bounds as was previously the case) for the number of generalized diagonals, etc. in polygonal billiards.

There are many other instances where some problem may be naturally (re)formulated in terms of translation surfaces coming from polygonal billiards; then the magic wand implies additional structure on a relevant family of translation surfaces, which yields insight into the original problem. Below we outline two concrete examples of this:

### The illumination problem

Given a room, how many light-bulbs are required to light it? Or, to abstract the problem a little: given a polygonal domain P (or really any planar domain, but let’s stick to polygons for now) and a point $x \in P$, which points in P can (or cannot) be reached by billiard trajectories through x? A point y which can be reached from x is said to be  illuminated from x.

Billiard trajectories very much resemble light-ray trajectories (at least locally)—indeed the word “optical” appeared in our description of billiard systems—and so it should be no surprise that the study of billiard systems and hence of translation surfaces yields insight into this and related problems. Indeed, as this wonderfully-named paper notes, the illumination problem “elementary properties which can be fruitfully studied using the dynamical behavior of the $\mathrm{SL}(2,\mathbb{R})$-action on the moduli space of translation surfaces.”

Using the magic wand theorem, and that the geometric properties considered in the illumination problem produce closed sets of the moduli space $\mathcal{H}$Lelièvre-Monteil-Weiss have proved that, for any P and any $x \in P$, there are finitely many $y \in P$ which are not illuminated from x.

### The wind-tree model

The wind-tree model was originally formulated by statistical physicists Paul and Tatiana Ehrenfest as a model for a Lorenz gas: in this model, particles (the “wind”) travel in straight-line trajectories in the plane $\mathbb{R}^2$, reflecting off rectangular obstacles (“trees”) placed along a $\mathbb{Z}^2$ lattice in a billiards-like fashion. One can also describe it, precisely, as billiards in the plane with these rectangles removed.

One can form a translation surface by restricting to some suitable subset of the plane and obstacles, and gluing the sides together: (figure taken from the also wonderfully-named “Cries and whispers in wind-tree forests“)

The result is a genus-5 flat surface in the stratum $\mathcal{H}(2^4)$.

One can then describe the behavior of the trajectories in terms of properties of the translation surface, e.g. Delecroix-Hubert-Lelièvre have computed the diffusion of divergent trajectories, for rectangular obstacles of any size, in terms of the Lyapunov exponents of  a natural dynamical system (the Kontsevich-Zorich cocycle) on a certain stratum of genus-5 translation surfaces—not the one specified above, but a quotient thereof.

## References

Alex Wright’s article describes Eskin-Mirzakhani-Mohammadi result, the context for it, as well as applications and connections to nearby areas.

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# Translation surfaces and friends

Recall that we earlier defined translation surfaces as maximal atlases of coordinate charts into the Euclidean plane $\mathbb{R}^2$, with finitely many cone singularities and where the transition maps between coordinate charts are translations in $\mathbb{R}^2$.

Recall that we were led to this notion starting from polygonal billiards, but not every translation surface, generally defined, comes from a rational polygonal billiard.

## Enter Riemann surfaces and moduli spaces

A translation surface may also be thought of as a Riemann surface, together with an associated holomorphic 1-form—a continuous choice of specified direction (“up”) at every point; such a pair of structures canonically defines a (singular) flat structure on the surface, with a distinguished vertical direction.

To wit: we take the holomorphic 1-form $\omega$ to be (locally) our dz. Since $\omega$ is holomorphic, its zeroes are isolated: a zero of degree d corresponds exactly to a conical singularity with cone angle $2\pi(d+1)$. Where $\omega$ is not zero, it is represented (locally) by dz, whence we obtain a complex coordinate z, which in turn specifies (locally) a Euclidean structure. Since dz is globally well-defined on our translation surface, the resulting flat structure is also well-defined, away from the conical singularities.

Conversely, given a flat structure with conical singularities $P_1, \dots, P_n$, and a specified direction, consider a fundamental polygon $P_1P_2 \cdots P_n$ embedded (anywhere, but with orientation dictated by the specified direction) in the complex plane. The fundamental polygon inherits a natural complex coordinate z. This does not descend to the translation surface, but since the identification maps are all of the form $z \mapsto z + \zeta$ where $\zeta \in \mathbb{C}$ is a constant (for each identification map), the holomorphic 1-form dz does, and we obtain a holomorphic 1-form $\omega$ on our translation surface. $\omega$ is zero exactly at the conical singularities, with cone angle being related to degree as above.

The Riemann surface perspective emphasizes more clearly the presence of a moduli space of translation surfaces (on for a fixed genus g), which we presently define as a space of pairs $(X, \omega)$, where X denotes a Riemann surface structure, and $\omega$ a choice of holomorphic 1-form, modulo some natural equivalence relation. The 1-form specifies a distinguished direction—so there is a well-defined notion of “north” on the surface. Two such pairs $(X_1, \omega_1)$ and $(X_2, \omega_2)$ are considered equivalent if there is a conformal map from $X_1$ to $X_2$ which takes (specified) zeroes of $\omega_1$ to (specified) zeroes of $\omega_2$.

The moduli space is divided into distinct strata $\mathcal{H}(d_1, \dots, d_m)$, consisting of forms with zeroes of degree $d_1, \dots, d_m$ with $d_1 + \dots + d_m = 2g-2$. This last identity follows from the formula for the sum of degrees of zeroes of a holomorphic 1-form on a Riemann surface of genus g, and can be interpreted as a Gauss-Bonnet formula for the singular flat metric.

There is a $\mathrm{SL}_2\mathbb{R}$ action on this moduli space (or, really, on each stratum) which is most easily described back in the framework of flat geometry: given any pair $(X,\omega)$, build the corresponding flat polygon (with distinguished vertical direction.) Elements of $\mathrm{SL}_2\mathbb{R}$ act (linearly) on these flat polygons, thought of as being embedded in the Euclidean plane, with (say) some vertex at the origin. This action preserves parallelisms between edges; and hence is (induces) an action on the corresponding flat surfaces.

(figure from the Zorich survey—the $\mathrm{SL}(2,\mathbb{R})$ action, depicted on the left, changes the affine structure but not the translation structure; the “cut-and-paste” on the right changes the translation structure, but not the affine structure.)

We can study the dynamics of this action, and this turns out to be surprisingly (or perhaps not surprisingly—or maybe that is only with the benefit of giant-assisted hindsight) rich …

(There were already hints of this in the last post, when we talked about “changing the translation structure while preserving the affine structure” and how Masur used this idea to prove his theorem on counting closed geodesics.)

## Counting closed geodesics and saddle connections

As previously noted in the slightly more restricted context of rational polygonal billiards, directional flow on a flat surface is uniquely ergodic in almost every direction.

This is most transparently seen in the case of a torus: geodesics with rational slopes are closed, while those with irrational slopes are equidistributed. Geodesics on flat surfaces of higher genera exhibit certain similiarities: the closed geodesics also appear in parallel families, although in higher genera these do not fill the whole surface, but only flat cylinders with conical singularities on the boundary.

Related to closed geodesic are saddle connections, which are geodesic segments joining two conical singularities (which may coincide), with no conical points in their interior. On a flat torus there are no conical singularities, and so any closed loop can be tightened to a closed geodesic; on a more general flat surface, this tightening process can produce either a closed geodesic, or—if at some point in the process we hit one of the singularities on the surface, which, as one might imagine, is the more generic case—a union of saddle connections.

Masur and Eskin have found quadratic asymptotics, as a function of length, for the number of [cylindrical families of] closed geodesics—this was discussed more in the previous post—and the number of saddle connections. The constants which appear in these asymptotics are called the Siegel-Veech constants. There are also (somewhat surprising) quadratic asymptotics for multiple cylindrical families in the same parallel direction.

These results are interesting in their own right—on a [rational] billiard table, for instance, generalized diagonals (trajectories joining two of the corners, possibly after reflections) unfold to saddle connections and periodic trajectories unfold to closed regular geodesics—but are also useful for at least two other reasons:

One, degeneration of “configurations” of parallel saddle connections or closed geodesics leads us into cusped regions in the boundary of (strata in the) moduli space, and so a description of such configurations gives us a description of the cusps of our strata. Local considerations involving short / degenerating saddle connections also lead us to relations between and structural results about the strata, which can be described more carefully / analytically in the language of Abelian differentials.

Two, configurations of saddle connections and closed geodesics are also useful as invariants of $\mathrm{SL}(2,\mathbb{R})$ orbits—something that we will refer back to below.

## Volume of moduli space

We remark that this last counting problem can be related to computations of volumes of the moduli space, via the observation that the Siegel-Veech constants can be obtained as a limit of the form $\lim_{\epsilon \to 0} \frac{1}{\pi \epsilon^2} \frac{\mathrm{Vol}(\epsilon\mbox{-neighborhood of cusp }\mathcal{C})}{\mathrm{Vol} \mathcal{H}_1^\circ(d_1, \dots, d_n)}$ where $\mathcal{C}$ is a specified configuration of saddle connections or closed geodesics.

Athreya-Eskin-Zorich used this idea to obtain explicit formulas (conjectured by Kontsevich based on experimental evidence) for the volumes of strata in genus 0, by counting generalized diagonals and periodic trajectories on right-angled billiards. In general, of course, the relation between the two problems can be exploited in both directions: results on volumes of strata can also be used to obtain results on the counts of saddle connections / closed geodesics.

There are a number of alternative strategies for finding these volumes; the following is so far the most general:

The general idea is to simply use asymptotics for counts of integer lattice points, where the lattice is defined in terms of cohomological period coordinates. We can count the number of such points in a sphere or hyperboloid (which is the unit sphere for an indefinite quadratic form of suitable signature) to estimate its volume, and take a derivative to estimate the volume of the boundary hypersurface.

Integer lattice points may be thought of, geometrically, as flat surfaces tiled by square flat tori, and combinatorial geometric methods may be used to count these: in the simplest cases we can count directly; slightly more generally we can consider the graph with conical points as vertices and horizontal saddle connections as edges, leading to the notions of ribbon graphs and separatrix diagrams.

In the most general case we turn, following an idea of Eskin-Okounkov, to representation theory: suppose there are N squares; label them, and consider the permutation $\pi$ on [N] which sends j to the square $\pi(j)$ which we get to by starting at j and moving left, up, right, and down in turn. For the generic square j, $\pi$ fixes j, but near the conical points it acts non-trivially, and indeed it is a product of m cycles of lengths $(d_1+1), \dots, (d_m+1)$.

It then suffices to count the number of permutations of N with such a property … except there is a nontrivial correction needed to pick out only those permutations which correspond to connected square-tiled surfaces. Eskin-Okounkov-Pandharipande pushed through this strategy to obtain explicit quantitative results, with a strong arithmetic flavor. These results may be made explicit, although there is considerable computational work involved; by comparison, other approaches such as the work of Athreya-Eskin-Zorich referred to above can produce simpler formulas in special cases.

## Applications

We have described above and previously how translation surfaces are related to (rational) polygonal billiards, and how the former provide a powerful framework for the study of the latter. Here (and in a subsequent post) we present a number of other applications:

### Electron transport

S. P. Novikov suggested the following as a mathematical formulation of electron transport in metals: consider a periodic surface $\widetilde{M^2} \subset \mathbb{R}^3$; an affine plane in $\mathbb{R}^3$ intersects this in some union of closed and unbounded intervals. Question: how does an unbounded component propagate in $\mathbb{R}^3$ (as we move the affine plane in some continuous fashion?)

After we quotient by the period lattice (taken to be $\mathbb{Z}^3$), we are looking at plane sections of the quotient surface $M^2 \subset \mathbb{T}^3$. Our original intersection can be viewed as level curves of a linear function $f(x,y,z) = ax + by + cz$ restricted to $\widetilde{M^2}$, but this does not push down to the quotient; instead, we consider the codimension-one foliation of $M^2$ defined by the closed 1-form $df = a\,dx + b\,dy + c\,dz$.

Our question can then be reformulated as follows: what do lifts of leaves of this foliation on $M^2 \subset \mathbb{T}^3$ look like upstairs, in $\widetilde{M^2} \subset \mathbb{R}^3$?

Closed 1-forms on surfaces can be straightened to geodesic foliations in appropriate flat metrics iff any cycle formed from a union of closed paths following a sequence of saddle connections is homologically non-trivial. The surfaces and 1-forms obtained from Novikov’s problem can be modified (decomposed and surgered) to satisfy this criterion, and after these reductions we are left exactly in the world of flat structures with closed 1-forms on them, i.e. translation surfaces.

### Invisibility

In a subsequent post we describe the illumination problem; a related problem concerns invisibility—more precisely: whether a body with mirror surfaces can be “invisible” from some direction/s, because light rays travelling in these direction/s are reflected in such a way that they continue along trajectories in the same direction/s; or whether a body can be similarly made invisible in certain driections through the strategic placement of mirrors around it. It is a conjecture of Plakhov the set of directions that are invisible for any fixed body has measure zero. This conjecture is closely connected with the (similar) Ivrii conjecture on the measure of the set of periodic billiard trajectories in a bounded domain: if Ivrii’s conjecture is true then, most probably, true also is the conjecture on invisible light rays.

## Flat (but not very), and real affine …

There are at least two ways in which the notion of translation surfaces can be generalized: one is to consider flat structures with non-trivial linear holonomy, which “forces a generic geodesic to come back and to intersect itself again and again in different directions.” The other to consider real affine structures, which are maximal collections of charts on a closed surface where all of the transition maps are of the form $f(z) = az+b$ where $a > 0$ (and is in particular real) and $b \in \mathbb{C}$.

These remain rather less well-studied and more mysterious, or, in other words / from another perspective, present potentially rich sources of interesting open problems …

## References

Zorich’s survey covers a broad range of ideas, and contains many further references

Hubert-Masur-Schmidt-Zorich have a (slightly outdated) list of open problems on translation surfaces , from a conference at Luminy.

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# The many faces of the hyperbolic plane

$\mathbb{H}^2$ is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

1. It is diffeomorphic to $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ as a topological space (or, indeed, isometric as a Riemannian manifold, if we give $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ a left-invariant metric): to show this, we note that $\mathbb{H}^2$ has isometry group $\mathrm{SL}(2,\mathbb{R}) / \pm\mathrm{id}$, and the subgroup of isometries which stabilize any given $p \in \mathbb{H}^2$ is isomorphic to $\mathrm{SO}(2)$.
2. Since any positive-definite binary quadratic form is given by a symmetric 2-by-2 matrix with positive eigenvalues, and since the group of linear transformations on $\mathbb{R}^2$ preserving any such given form is isomorphic to $\mathrm{O}(2)$, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of positive-definite binary quadratic forms of determinant 1, via the map from $\mathrm{SL}(2,\mathbb{R})$ to the symmetric positive-definite 2-by-2 matrices given by $g \mapsto g^T g$.
3. $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2$ is also the moduli space of marked Riemann surfaces of genus 1, i.e. the Teichmüller space Teich(S) of the torus S. One way to prove this is to note that any such marked surface is the quotient of $\mathbb{R}^2$ by a $\mathbb{Z}^2$ action; after a suitable conformal transformation, we may assume that the generators of this $\mathbb{Z}^2$ act as $z \mapsto z + 1$ and $z \mapsto z + \tau$ for some $z \in \mathbb{H}^2$ (in the upper half-plane model.) But now $\tau$ is the unique invariant specifying this point in our moduli space.
4. Since any marked Riemann surface of genus 1 has a unique flat metric (inherited as a quotient manifold of the Euclidean plane), $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the moduli space of marked flat 2-tori of unit area.
5. Since there is a unique unit-covolume marked lattice associated to each marked complex torus in the above, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of marked lattices in $\mathbb{R}^2$ with unit covolume.
6. Note we may go directly between marked Riemann surfaces and quadratic forms by considering intersection forms.
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