Thurston boundary

Last updated

In mathematics the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. It can be interpreted as the space of projective measured foliations on the surface.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, the Teichmüller spaceT(S) of a (real) topological surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in T(S) may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from S to itself.

Boundary (topology) the dividing line or location between two areas / set of points in a topological space

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include bd(S), fr(S), and ∂S. Some authors use the term frontier instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology and manifold theory. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set ; that is, S \ S known alternatively as the residue of S. Other authors use the term boundary for those points that are in S but not in its interior.


The Thurston boundary of the Teichmüller space of a closed surface of genus is homeomorphic to a sphere of dimension . The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary.

In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

Measured foliations on surfaces

Let be a closed surface. A measured foliation on is a foliation on which may admit isolated singularities, together with a transverse measure, i.e. a function which to each arc transverse to the foliation associates a positive real number . The foliation and the measure must be compatible in the sense that the measure is invariant if the arc is deformed with endpoints staying in the same leaf. [1]


In mathematics, a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1. The number p is called the dimension of the foliation and q = n - p is called its codimension.

Let be the space of isotopy classes of closed simple curves on . A measured foliation can be used to define a function as follows: if is any curve let

where the supremum is taken over all collections of disjoint arcs which are transverse to (in particular if is a closed leaf of ). Then if the intersection number is defined by:


Two measured foliations are said to be equivalent if they define the same function on (there is a topological criterion for this equivalence via Whitehead moves). The space of projective measured laminations is the image of the set of measured laminations in the projective space via the embedding . If the genus of is at least 2, the space is homeomorphic to the -dimensional sphere (in the case of the torus it is the 2-sphere; there are no measured foliations on the sphere).

Compactification of Teichmüller space

Embedding in the space of functionals

Let be a closed surface. Recall that a point in the Teichmüller space is a pair where is an hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to ) and a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set of isotopy classes of simple closed curves on as follows. If and then is defined to be the length of the unique closed geodesic on in the isotopy class . The map is an embedding of into , which can be used to give the Teichmüller space a topology (the right-hand side being given the product topology).

In fact, the map to the projective space is still an embedding: let denote the image of there. Since this space is compact, the closure is compact: it is called the Thurston compactification of the Teichmüller space.

The Thurston boundary

The boundary is equal to the subset of . The proof also implies that the Thurston compactfification is homeomorphic to the -dimensional closed ball. [2]


Pseudo-Anosov diffeomorphisms

A diffeomorphism is called pseudo-Anosov if there exists two transverse measured foliations, such that under its action the underlying foliations are preserved, and the measures are multiplied by a factor respectively for some (called the stretch factor). Using his compactification Thurston proved the following characterisation of pseudo-Anosov mapping classes (i.e. mapping classes which contain a pseudo-Anosov element), which was in essence known to Nielse and is usually called the Nielsen-Thurston classification. A mapping class is pseudo-Anosov if and only if:

In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface.

The proof relies on the Brouwer fixed point theorem applied to the action of on the Thurston compactification . If the fixed point is in the interior then the class is of finite order; if it is on the boundary and the underlying foliation has a closed leaf then it is reducible; in the remaining case it is possible to show that there is another fixed point corresponding to a transverse measured foliation, and to deduce the pseudo-Anosov property.

Applications to the mapping class group

The action of the mapping class group of the surface on the Teichmüller space extends continuously to the Thurston compactification. This provides a powerful tool to study the structure of this group; for example it is used in the proof of the Tits alternative for the mapping class group. It can also be used to prove various results about the subgroup structure of the mapping class group. [3]

Applications to 3manifolds

The compactification of Teichmüller space by adding measured foliations is essential in the definition of the ending laminations of an hyperbolic 3-manifold.

Actions on real trees

A point in Teichmüller space can alternatively be seen as a faithful representation of the fundamental group into the isometry group of the hyperbolic plane , up to conjugation. Such an isometric action gives rise (via the choice of a principal ultrafilter) to an action on the asymptotic cone of , which is a real tree. Two such action are equivariantly isometric if and only if they come from the same point in Teichmüller space. The space of such actions (endowed with a natural topology) is compact, and hence we get another compactification of Teichmüller space. A theorem of R. Skora states that this compactification is equivariantly homeomorphic the Thurston compactification. [4]


  1. Fathi, Laudenbach & Poenaru 2012, Exposé 5.
  2. Fathi, Laudenbach & Poenaru 2012, Exposé 8.
  3. Ivanov 1992.
  4. Bestvina, Mladen. "-trees in topology, geometry and group theory". Handbook of geometric topology. North-Holland. pp. 55–91.

Related Research Articles

In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.

Einstein–Hilbert action

The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the principle of least action. With the (− + + +) metric signature, the gravitational part of the action is given as

Einstein tensor Tensor used in general relativity

In differential geometry, the Einstein tensor is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner consistent with energy and momentum conservation.

Mathematics of general relativity

The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

Maxwells equations in curved spacetime

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the Central Limit Theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables of order 1, then X is of order and its law is approximately Gaussian.

Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst % of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio

Normal-inverse-gamma distribution

In probability theory and statistics, the normal-inverse-gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

Lovelock theory of gravity

In theoretical physics, Lovelock's theory of gravity is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D. In this sense, Lovelock's theory is the natural generalization of Einstein's General Relativity to higher dimensions. In three and four dimensions, Lovelock's theory coincides with Einstein's theory, but in higher dimensions the theories are different. In fact, for D > 4 Einstein gravity can be thought of as a particular case of Lovelock gravity since the Einstein–Hilbert action is one of several terms that constitute the Lovelock action.

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.

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields. It is also the modern name for what used to be called the absolute differential calculus, developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.

Affine gauge theory is classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold . For instance, these are gauge theory of dislocations in continuous media when , the generalization of metric-affine gravitation theory when is a world manifold and, in particular, gauge theory of the fifth force.

Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

Loop representation in gauge theories and quantum gravity

Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states. The idea is well known in the context of lattice Yang–Mills theory. Attempts to explore the continuous loop representation was made by Gambini and Trias for canonical Yang–Mills theory, however there were difficulties as they represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical framework to treat gauge theories and quantum gravity in terms of their fundamental physical excitations.

In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary it is in general a purely measure-theoretical object and not a boundary in the topological sense. However, in the case where the random walk is on a topological space the Poisson boundary can be related to the Martin boundary which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to harmonic functions on the space via generalisations of the Poisson formula.