In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
a translation surface is a surface obtained from identifying the sides of a polygon in the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
by translations. An equivalent definition is a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
together with a
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
.
These surfaces arise in
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s where they can be used to model
billiards
Cue sports are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as .
There are three major subdivisions of ...
, and in
Teichmüller theory
Teichmüller is a German surname (German for ''pond miller'') and may refer to:
* Anna Teichmüller (1861–1940), German composer
* :de:Frank Teichmüller (19?? – now), former German IG Metall district manager "coast"
* Gustav Teichmüller (183 ...
. A particularly interesting subclass is that of Veech surfaces (named after
William A. Veech
William A. Veech was the Edgar O. Lovett Professor of Mathematics at Rice University[Faculty pro ...](_blank)
) 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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
on
, and form the broken lines
and
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 In mathematics, a branched covering is a map that is almost a covering map, except on a small set.
In topology
In topology, a map is a ''branched covering'' if it is a covering map everywhere except for a nowhere dense set known as the branch set. ...
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
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
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
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
from
Girard's theorem
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
.
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
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
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
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
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
:
where
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
. The real projectivization admits a natural section
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
William A. Veech was the Edgar O. Lovett Professor of Mathematics at Rice University[Faculty pro ...](_blank)
and
Howard Masur
Howard Alan Masur is an American mathematician who works on topology, geometry and combinatorial group theory.
Biography
Masur was an invited speaker at the 1994 International Congress of Mathematicians in Zürich. and is a fellow of the Amer ...
.
In the 90's
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
and
Anton Zorich
Anton V. Zorich (in Russian: ''Антон Владимирович Зорич''; born 3 September 1962) is a Russian mathematician at the Institut de mathématiques de Jussieu. He is the son of Vladimir A. Zorich. He received his Ph.D. from Mosco ...
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
Andrei Yuryevich Okounkov (russian: Андре́й Ю́рьевич Окунько́в, ''Andrej Okun'kov'') (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematic ...
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.
More recently Chen, Möller, Sauvaget, and
don Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Col ...
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.
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 In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of ho ...
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 mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.
Intuitively, let ''f'' : ''D' ...
s 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
, 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, though almost all such rays do so.
Veech surfaces
The Veech group
If
is a translation surface its ''Veech group'' is the
Fuchsian group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...
which is the image in
of the subgroup
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.
Veech surfaces
A Veech surface is by definition a translation surface whose Veech group is a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
in
, equivalently its action on the
hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
admits a
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
of finite volume. Since it is not cocompact it must then contain parabolic elements.
Examples of Veech surfaces are the square-tiled surfaces, whose Veech groups are
commensurable to the
modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
. 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
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
that is
totally real
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...
.
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
(or
in the case of a half-translation surface
). The ''geodesic flow'' on
with direction
is the
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psych ...
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 mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
. 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
. 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
:
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:
:
,
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.
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
In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
as follows:
is equal to
where
for
. Then this map is an
interval exchange transformation
In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and ...
and it can be used to study the dynamic of the geodesic flow.
Notes
References
*
*
*
*
*{{cite book , last=Zorich , first=Anton , chapter=Flat surfaces , title=Frontiers in Number Theory, Physics and Geometry. Volume 1: On random matrices, zeta functions and dynamical systems , editor-last1=Cartier , editor-first1=P. , editor-last2=Julia , editor-first2=B. , editor-last3=Moussa , editor-first3=P. , editor-last4=Vanhove , editor-first4=P. , publisher=Springer-Verlag , year=2006 , arxiv=math/0609392 , bibcode=2006math......9392Z
Surfaces
Dynamical systems