# 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. The Thurston boundary can be interpreted as the space of projective measured foliations on the surface.

## Contents

The Thurston boundary of the Teichmüller space of a closed surface of genus ${\displaystyle g}$ is homeomorphic to a sphere of dimension ${\displaystyle 6g-7}$. The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary.

## Measured foliations on surfaces

Let ${\displaystyle S}$ be a closed surface. A measured foliation${\displaystyle ({\mathcal {F}},\mu )}$ on ${\displaystyle S}$ is a foliation ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle S}$ which may admit isolated singularities, together with a transverse measure${\displaystyle \mu }$, i.e. a function which to each arc ${\displaystyle \alpha }$ transverse to the foliation ${\displaystyle {\mathcal {F}}}$ associates a positive real number ${\displaystyle \mu (\alpha )}$. 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]

Let ${\displaystyle {\mathcal {S}}}$ be the space of isotopy classes of closed simple curves on ${\displaystyle S}$. A measured foliation ${\displaystyle ({\mathcal {F}},\mu )}$ can be used to define a function ${\displaystyle i(({\mathcal {F}},\mu ),\cdot )\in \mathbb {R} _{+}^{\mathcal {S}}}$ as follows: if ${\displaystyle \gamma }$ is any curve let

${\displaystyle \mu (\gamma )=\sup _{\alpha _{1},\ldots ,\alpha _{r}}\left(\sum _{i=1}^{r}\mu (\alpha _{i})\right)}$

where the supremum is taken over all collections of disjoint arcs ${\displaystyle \alpha _{1}\ldots ,\alpha _{r}\subset \gamma }$ which are transverse to ${\displaystyle {\mathcal {F}}}$ (in particular ${\displaystyle \mu (\gamma )=0}$ if ${\displaystyle \gamma }$ is a closed leaf of ${\displaystyle {\mathcal {F}}}$). Then if ${\displaystyle \sigma \in {\mathcal {S}}}$ the intersection number is defined by:

${\displaystyle i{\bigl (}({\mathcal {F}},\mu ),\sigma {\bigr )}=\inf _{\gamma \in \sigma }\mu (\gamma )}$.

Two measured foliations are said to be equivalent if they define the same function on ${\displaystyle {\mathcal {S}}}$ (there is a topological criterion for this equivalence via Whitehead moves). The space ${\displaystyle {\mathcal {PMF}}}$ of projective measured laminations is the image of the set of measured laminations in the projective space ${\displaystyle \mathbb {P} (\mathbb {R} _{+}^{\mathcal {S}})}$ via the embedding ${\displaystyle i}$. If the genus ${\displaystyle g}$ of ${\displaystyle S}$ is at least 2, the space ${\displaystyle {\mathcal {PMF}}}$ is homeomorphic to the ${\displaystyle 6g-7}$-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 ${\displaystyle S}$ be a closed surface. Recall that a point in the Teichmüller space is a pair ${\displaystyle (X,f)}$ where ${\displaystyle X}$ is a hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to ${\displaystyle -1}$) and ${\displaystyle f}$ a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set ${\displaystyle {\mathcal {S}}}$ of isotopy classes of simple closed curves on ${\displaystyle {\mathcal {S}}}$ as follows. If ${\displaystyle x=(X,f)\in T(S)}$ and ${\displaystyle \sigma \in {\mathcal {S}}}$ then ${\displaystyle \ell (x,\sigma )}$ is defined to be the length of the unique closed geodesic on ${\displaystyle X}$ in the isotopy class ${\displaystyle f_{*}\sigma }$. The map ${\displaystyle x\mapsto \ell (x,\cdot )}$ is an embedding of ${\displaystyle T(S)}$ into ${\displaystyle \mathbb {R} _{+}^{\mathcal {S}}}$, 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 ${\displaystyle \mathbb {P} (\mathbb {R} _{+}^{\mathcal {S}})}$ is still an embedding: let ${\displaystyle {\mathcal {T}}}$ denote the image of ${\displaystyle T(S)}$ there. Since this space is compact, the closure ${\displaystyle {\overline {\mathcal {T}}}}$ is compact: it is called the Thurston compactification of the Teichmüller space.

### The Thurston boundary

The boundary ${\displaystyle {\overline {\mathcal {T}}}\setminus {\mathcal {T}}}$ is equal to the subset ${\displaystyle {\mathcal {PMF}}}$ of ${\displaystyle \mathbb {P} (\mathbb {R} _{+}^{\mathcal {S}})}$. The proof also implies that the Thurston compactfification is homeomorphic to the ${\displaystyle 6g-6}$-dimensional closed ball. [2]

## Applications

### Pseudo-Anosov diffeomorphisms

A diffeomorphism ${\displaystyle S\to S}$ 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 ${\displaystyle \lambda ,\lambda ^{-1}}$ respectively for some ${\displaystyle \lambda >1}$ (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 Nielsen and is usually called the Nielsen-Thurston classification. A mapping class ${\displaystyle \phi }$ is pseudo-Anosov if and only if:

• it is not reducible (i.e. there is no ${\displaystyle k\geq 1}$ and ${\displaystyle \sigma \in {\mathcal {S}}}$ such that ${\displaystyle (\phi ^{k})_{*}\sigma =\sigma }$);
• it is not of finite order (i.e. there is no ${\displaystyle k\geq 1}$ such that ${\displaystyle \phi ^{k}}$ is the isotopy class of the identity).

The proof relies on the Brouwer fixed point theorem applied to the action of ${\displaystyle \phi }$ on the Thurston compactification ${\displaystyle {\overline {\mathcal {T}}}}$. 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 ${\displaystyle S}$ 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 3–manifolds

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

## Actions on real trees

A point in Teichmüller space ${\displaystyle T(S)}$ can alternatively be seen as a faithful representation of the fundamental group ${\displaystyle \pi _{1}(S)}$ into the isometry group ${\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}$ of the hyperbolic plane ${\displaystyle \mathbb {H} ^{2}}$, up to conjugation. Such an isometric action gives rise (via the choice of a principal ultrafilter) to an action on the asymptotic cone of ${\displaystyle \mathbb {H} ^{2}}$, which is a real tree. Two such actions 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]

## Notes

1. Fathi, Laudenbach & Poénaru 2012, Exposé 5.
2. Fathi, Laudenbach & Poénaru 2012, Exposé 8.
3. Bestvina, Mladen. "${\displaystyle \mathbb {R} }$-trees in topology, geometry and group theory". Handbook of geometric topology. North-Holland. pp. 55–91.

## Related Research Articles

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices that are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In mathematical analysis and in probability theory, a σ-algebra on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space.

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-12 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra.

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 = np is called its codimension.

In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. 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 mathematics, the Teichmüller space of a (real) topological surface is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

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.

In mathematical physics, the gamma matrices, also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Dirac in 1928.

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, cylinder set measure is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space.

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.

In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space . Non-singular quintic threefolds are Calabi–Yau manifolds.

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.

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.

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.

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.

The Fréchet inception distance (FID) is a metric used to assess the quality of images created by a generative model, like a generative adversarial network (GAN). Unlike the earlier inception score (IS), which evaluates only the distribution of generated images, the FID compares the distribution of generated images with the distribution of a set of real images.

## References

• Fathi, Albert; Laudenbach, François; Poénaru, Valentin (2012). Thurston's work on surfaces Translated from the 1979 French original by Djun M. Kim and Dan Margalit. Mathematical Notes. Vol. 48. Princeton University Press. pp. xvi+254. ISBN   978-0-691-14735-2.
• Ivanov, Nikolai (1992). Subgroups of Teichmüller Modular Groups. American Math. Soc.