Translation surface explained
In mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form.
These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory. A particularly interesting subclass is that of Veech surfaces (named after William A. Veech) which are the most symmetric ones.
Definitions
Geometric definition
A translation surface is the space obtained by identifying pairwise by translations the sides of a collection of plane polygons.
Here is a more formal definition. Let
be a collection of (not necessarily convex) polygons in the Euclidean plane and suppose that for every side
of any
there is a side
of some
with
and
for some nonzero vector
(and so that
. Consider the space obtained by identifying all
with their corresponding
through the map
.
The canonical way to construct such a surface is as follows: start with vectors
and a
permutation
on
, and form the broken lines
L=x,x+\vecw1,\ldots,x+\vecw1+ … +\vecwn
and
L'=x,x+\vecw\sigma(1),\ldots,x+\vecw\sigma(1)+ … +\vecw\sigma(n)
starting at an arbitrarily chosen point. In the case where these two lines form a polygon (i.e. they do not intersect outside of their endpoints) there is a natural side-pairing.
The quotient space is a closed surface. It has a flat metric outside the set
images of the vertices. At a point in
the sum of the angles of the polygons around the vertices which map to it is a positive multiple of
, and the metric is singular unless the angle is exactly
.
Analytic definition
Let
be a translation surface as defined above and
the set of singular points. Identifying the Euclidean plane with the complex plane one gets coordinates charts on
with values in
. Moreover, the changes of charts are holomorphic maps, more precisely maps of the form
for some
. This gives
the structure of a Riemann surface, which extends to the entire surface
by Riemann's theorem on
removable singularities. In addition, the differential
where
is any chart defined above, does not depend on the chart. Thus these differentials defined on chart domains glue together to give a well-defined holomorphic 1-form
on
. The vertices of the polygon where the cone angles are not equal to
are zeroes of
(a cone angle of
corresponds to a zero of order
).
In the other direction, given a pair
where
is a compact Riemann surface and
a holomorphic 1-form one can construct a polygon by using the complex numbers
where
are disjoint paths between the zeroes of
which form an integral basis for the relative cohomology.
Examples
The simplest example of a translation surface is obtained by gluing the opposite sides of a parallelogram. It is a flat torus with no singularities.
If
is a regular
-gon then the translation surface obtained by gluing opposite sides is of genus
with a single singular point, with angle
.
If
is obtained by putting side to side a collection of copies of the unit square then any translation surface obtained from
is called a
square-tiled surface. The map from the surface to the flat torus obtained by identifying all squares is a
branched covering with branch points the singularities (the cone angle at a singularity is proportional to the degree of branching).
Riemann - Roch and Gauss - Bonnet
Suppose that the surface
is a closed Riemann surface of genus
and that
is a nonzero holomorphic 1-form on
, with zeroes of order
. Then the
Riemann–Roch theorem implies that
If the translation surface
is represented by a polygon
then triangulating it and summing angles over all vertices allows to recover the formula above (using the relation between cone angles and order of zeroes), in the same manner as in the proof of the
Gauss–Bonnet formula for hyperbolic surfaces or the proof of
Euler's formula from Girard's theorem.
Translation surfaces as foliated surfaces
If
is a translation surface there is a natural
measured foliation on
. If it is obtained from a polygon it is just the image of vertical lines, and the measure of an arc is just the euclidean length of the horizontal segment homotopic to the arc. The foliation is also obtained by the level lines of the imaginary part of a (local) primitive for
and the measure is obtained by integrating the real part.
Moduli spaces
Strata
Let
be the set of translation surfaces of genus
(where two such
are considered the same if there exists a holomorphic diffeomorphism
such that
). Let
be the
moduli space of Riemann surfaces of genus
; there is a natural map
mapping a translation surface to the underlying Riemann surface. This turns
into a locally trivial
fiber bundle over the moduli space.
To a compact translation surface
there is associated the data
where
are the orders of the zeroes of
. If
is any
integer partition of
then the stratum
is the subset of
of translation surfaces which have a holomorphic form whose zeroes match the partition.
The stratum
is naturally a complex orbifold of complex dimension
(note that
is the moduli space of tori, which is well-known to be an orbifold; in higher genus, the failure to be a manifold is even more dramatic). Local coordinates are given by
(X,\omega)\mapsto
\omega,\ldots,
\omega\right)
where
n=\dim(H1(S,\{x1,\ldots,xm\}))=2g+m-1
and
is as above a symplectic basis of this space.
Masur-Veech volumes
The stratum
admits a
-action and thus a real and complex projectivization
{lH(\alpha)}\to{lH}1(\alpha)\to{lH}2(\alpha)
. The real projectivization admits a natural section
{lH}1(\alpha)\to{lH}(\alpha)
if we define it as the space of translation surfaces of area 1.
The existence of the above period coordinates allows to endow the stratum
with an integral affine structure and thus a natural volume form
. We also get a volume form
on
by disintegration of
. The Masur-Veech volume
is the total volume of
for
. This volume was proved to be finite independently by
William A. Veech[1] and
Howard Masur.
[2] In the 90's Maxim Kontsevich and Anton Zorich evaluated these volumes numerically by counting the lattice points of
. They observed that
should be of the form
times a rational number. From this observation they expected the existence of a formula expressing the volumes in terms of intersection numbers on moduli spaces of curves.
Alex Eskin and Andrei Okounkov gave the first algorithm to compute these volumes. They showed that the generating series of these numbers are q-expansions of computable quasi-modular forms. Using this algorithm they could confirm the numerical observation of Kontsevich and Zorich.[3]
More recently Chen, Möller, Sauvaget, and don Zagier showed that the volumes can be computed as intersection numbers on an algebraic compactification of
. Currently the problem is still open to extend this formula to strata of half-translation surfaces.
[4] The SL2(R)-action
If
is a translation surface obtained by identifying the faces of a polygon
and
then the translation surface
is that associated to the polygon
. This defined a continuous action of
on the moduli space
which preserves the strata
. This action descends to an action on
that is ergodic with respect to
.
Half-translation surfaces
Definitions
A half-translation surface is defined similarly to a translation surface but allowing the gluing maps to have a nontrivial linear part which is a half turn. Formally, a translation surface is defined geometrically by taking a collection of polygons in the Euclidean plane and identifying faces by maps of the form
(a "half-translation"). Note that a face can be identified with itself. The geometric structure obtained in this way is a flat metric outside of a finite number of singular points with cone angles positive multiples of
.
As in the case of translation surfaces there is an analytic interpretation: a half-translation surface can be interpreted as a pair
where
is a Riemann surface and
a
quadratic differential on
. To pass from the geometric picture to the analytic picture one simply takes the quadratic differential defined locally by
(which is invariant under half-translations), and for the other direction one takes the Riemannian metric induced by
, which is smooth and flat outside of the zeros of
.
Relation with Teichmüller geometry
If
is a Riemann surface then the vector space of quadratic differentials on
is naturally identified with the tangent space to Teichmüller space at any point above
. This can be proven by analytic means using the
Bers embedding. Half-translation surfaces can be used to give a more geometric interpretation of this: if
are two points in Teichmüller space then by Teichmüller's mapping theorem there exists two polygons
whose faces can be identified by half-translations to give flat surfaces with underlying Riemann surfaces isomorphic to
respectively, and an affine map
of the plane sending
to
which has the smallest distortion among the
quasiconformal mappings in its isotopy class, and which is isotopic to
.
Everything is determined uniquely up to scaling if we ask that
be of the form
, where
ft:(x,y)\mapsto(etx,e-ty)
, for some
; we denote by
the Riemann surface obtained from the polygon
. Now the path
in Teichmüller space joins
to
, and differentiating it at
gives a vector in the tangent space; since
was arbitrary we obtain a bijection.
In facts the paths used in this construction are Teichmüller geodesics. An interesting fact is that while the geodesic ray associated to a flat surface corresponds to a measured foliation, and thus the directions in tangent space are identified with the Thurston boundary, the Teichmüller geodesic ray associated to a flat surface does not always converge to the corresponding point on the boundary,[5] though almost all such rays do so.[6]
Veech surfaces
The Veech group
If
is a translation surface its
Veech group is the
Fuchsian group which is the image in
of the subgroup
SL(X,\omega)\subsetSL2(R)
of transformations
such that
is isomorphic (as a translation surface) to
. Equivalently,
is the group of derivatives of affine diffeomorphisms
(where affine is defined locally outside the singularities, with respect to the affine structure induced by the translation structure). Veech groups have the following properties:
- They are discrete subgroups in
;
- They are never cocompact.
Veech groups can be either finitely generated or not.[7]
Veech surfaces
A Veech surface is by definition a translation surface whose Veech group is a lattice in
, equivalently its action on the
hyperbolic plane admits a
fundamental domain of finite volume. Since it is not cocompact it must then contain parabolic elements.
. The square can be replaced by any parallelogram (the translation surfaces obtained are exactly those obtained as ramified covers of a flat torus). In fact the Veech group is arithmetic (which amounts to it being commensurable to the modular group) if and only if the surface is tiled by parallelograms.
There exists Veech surfaces whose Veech group is not arithmetic, for example the surface obtained from two regular pentagons glued along an edge: in this case the Veech group is a non-arithmetic Hecke triangle group. On the other hand, there are still some arithmetic constraints on the Veech group of a Veech surface: for example its trace field is a number field that is totally real.[8]
Geodesic flow on translation surfaces
Geodesics
A geodesic in a translation surface (or a half-translation surface) is a parametrised curve which is, outside of the singular points, locally the image of a straight line in Euclidean space parametrised by arclength. If a geodesic arrives at a singularity it is required to stop there. Thus a maximal geodesic is a curve defined on a closed interval, which is the whole real line if it does not meet any singular point. A geodesic is closed or periodic if its image is compact, in which case it is either a circle if it does not meet any singularity, or an arc between two (possibly equal) singularities. In the latter case the geodesic is called a saddle connection.
If
(or
in the case of a half-translation surface) then the geodesics with direction theta are well-defined on
: they are those curves
which satisfy
\omega(\overset{ ⋅ }{c})=ei\theta
(or
\phi(\overset{ ⋅ }{c})=ei\theta
in the case of a half-translation surface
). The
geodesic flow on
with direction
is the
flow
on
where
is the geodesic starting at
with direction
if
is not singular.
Dynamical properties
On a flat torus the geodesic flow in a given direction has the property that it is either periodic or ergodic. In general this is not true: there may be directions in which the flow is minimal (meaning every orbit is dense in the surface) but not ergodic. On the other hand, on a compact translation surface the flow retains from the simplest case of the flat torus the property that it is ergodic in almost every direction.
Another natural question is to establish asymptotic estimates for the number of closed geodesics or saddle connections of a given length. On a flat torus
there are no saddle connections and the number of closed geodesics of length
is equivalent to
L2/\operatorname{volume}(T)
. In general one can only obtain bounds: if
is a compact translation surface of genus
then there exists constants (depending only on the genus)
such that the both
of closed geodesics and
of saddle connections of length
satisfy
(X,\omega)}\leNcg(L),Nsc(L)\le
(X,\omega)}.
Restraining to a probabilistic results it is possible to get better estimates: given a genus
, a partition
of
and a connected component
of the stratum
there exists constants
such that for almost every
the asymptotic equivalent holds:
Ncg(L)\sim
| ccgL2 |
\operatorname{volume |
(X,\omega)}
,
Nsc(L)\sim
| cscL2 |
\operatorname{volume |
(X,\omega)}.
The constants
are called
Siegel - Veech constants. Using the ergodicity of the
-action on
, it was shown that these constants can explicitly be computed as ratios of certain Masur-Veech volumes.
[9] Veech dichotomy
The geodesic flow on a Veech surface is much better behaved than in general. This is expressed via the following result, called the Veech dichotomy:
Let
be a Veech surface and
a direction. Then either all trajectories defied over
are periodic or the flow in the direction
is ergodic.
Relation with billiards
If
is a polygon in the Euclidean plane and
a direction there is a continuous dynamical system called a
billiard. The trajectory of a point inside the polygon is defined as follows: as long as it does not touch the boundary it proceeds in a straight line at unit speed; when it touches the interior of an edge it bounces back (i.e. its direction changes with an orthogonal reflection in the perpendicular of the edge), and when it touches a vertex it stops.
This dynamical system is equivalent to the geodesic flow on a flat surface: just double the polygon along the edges and put a flat metric everywhere but at the vertices, which become singular points with cone angle twice the angle of the polygon at the corresponding vertex. This surface is not a translation surface or a half-translation surface, but in some cases it is related to one. Namely, if all angles of the polygon
are rational multiples of
there is ramified cover of this surface which is a translation surface, which can be constructed from a union of copies of
. The dynamics of the billiard flow can then be studied through the geodesic flow on the translation surface.
For example, the billiard in a square is related in this way to the billiard on the flat torus constructed from four copies of the square; the billiard in an equilateral triangle gives rise to the flat torus constructed from an hexagon. The billiard in a "L" shape constructed from squares is related to the geodesic flow on a square-tiled surface; the billiard in the triangle with angles
is related to the Veech surface constructed from two regular pentagons constructed above.
Relation with interval exchange transformations
Let
be a translation surface and
a direction, and let
be the geodesic flow on
with direction
. Let
be a geodesic segment in the direction orthogonal to
, and defined the first recurrence, or
Poincaré map
as follows:
is equal to
where
for
. Then this map is an
interval exchange transformation and it can be used to study the dynamic of the geodesic flow.
References
- Book: Zorich, Anton . Flat surfaces . Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems . Cartier . P. . Julia . B. . Moussa . P. . Vanhove . P. . Springer-Verlag . 2006 . math/0609392 . 2006math......9392Z .
Notes and References
- gaussmeas. Veech . William A. . Gauss Measures for Transformations on the Space of Interval Exchange Maps . Annals of Mathematics . 115 . 2 . 1982 . 201 - 242 . 10.2307/1971391 . 1971391 .
- measfol. Masur . Howard . Interval Exchange Transformations and Measured Foliations . Annals of Mathematics . 115 . 1 . 1982 . 169 - 200 . 10.2307/1971341 . 1971341 .
- asy. Eskin . Alex . Okounkov . Andrei. Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials . Inventiones Mathematicae . 145 . 2001 . 1 . 59 - 103 . 10.1007/s002220100142. math/0006171 . 2001InMat.145...59E . 14125769 .
- intersection. Chen . Dawei . Möller . Martin. Sauvaget . Adrien . Zagier . Don Bernhard. Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials. Inventiones Mathematicae . 2019 . 222 . 1 . 283 . 10.1007/s00222-020-00969-4 . 1901.01785. 2020InMat.222..283C . 119655348 .
- Lenzhen . Anna . Teichmüller geodesics that do not have a limit in PMF . Geometry and Topology . 12 . 2008 . 177 - 197 . 10.2140/gt.2008.12.177. math/0511001 . 16047629 .
- Masur . Howard . Two boundaries of TeichmÛller space . Duke Math. J. . 49 . 1982 . 183 - 190 . 650376 . 10.1215/s0012-7094-82-04912-2.
- McMullen . Curtis T. . Teichmüller geodesics of infinite complexity . Acta Math. . 191 . 2 . 2003 . 191 - 223 . 10.1007/bf02392964. free .
- Hubert . Pascal . Lanneau . Erwan . Veech groups without parabolic elements . Duke Mathematical Journal . 2006 . 133 . 2 . 335–346 . 10.1215/s0012-7094-06-13326-4 . math/0503047 . 14274833 .
- principal . Eskin . Alex . Masur . Howard . Zorich. Anton. Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants . Publications Mathématiques de l'IHÉS . 2003 . 97 . 61–179 . 10.1007/s10240-003-0015-1 . math/0202134 . 119713402 .