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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.