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In [[mathematics]] the '''Thurston boundary''' of [[Teichmüller space]] of a surface is obtained as the [[Boundary (topology)|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 foliation]]s on the surface.
The Thurston boundary of the Teichmüller space of a closed surface of genus <math>g</math> is homeomorphic to a sphere of dimension <math>6g-7</math>. The action of the [[Mapping class group of a surface|mapping class group]] on the Teichmüller space extends continuously over the union with the boundary.
== Measured foliations on surfaces ==
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=== Embedding in the space of functionals ===
Let <math>S</math> be a closed surface. Recall that a point in the Teichmüller space is a pair <math>(X, f)</math> where <math>X</math> is an hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to <math>-1</math>) and <math>f</math> an homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set <math>\mathcal S</math> of isotopy classes of simple closed curves on <math>\mathcal S</math> as follows. If <math>x = (X, f) \in T(S)</math> and <math>\sigma \in \mathcal S</math> then <math>\ell(x, \sigma)</math> is defined to be the length of the unique closed geodesic on <math>X</math> in the isotopy class <math>f_*\sigma</math>. The map <math>x \mapsto \ell(x, \cdot)</math> is an embedding of <math>T(S)</math> into <math>\mathbb R_+^{\mathcal S}</math>, 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 <math>\mathbb P(\mathbb R_+^{\mathcal S})</math> is still an embedding: let <math>\mathcal T</matH> denote the image of <math>T(S)</math> there. Since this space is compact, the closure <math>\overline \mathcal T</math> is compact: it is called the ''Thurston compactification'' of the Teichmüller space.
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*it is not reducible (i.e. there is no <math>k \ge 1</math> and <math>\sigma \in \mathcal S</math> such that <math>(\phi^k)_*\sigma = \sigma</math>);
*it is not of finite order (i.e. there is no <math>k \ge 1</math> such that <math>\phi^k</math> is the isotopy class of the identity).
The proof relies on the [[Brouwer fixed point theorem]]applied to the action of <math>\phi</math> on the Thurston compactification <math>\overline \mathcal T</math>. 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 a surface|mapping class group]] of the surface <math>S</math> 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.{{sfn|Ivanov|1992}}
=== Applications to 3–manifolds ===
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*{{cite book | ref=harv | last1=Fathi | first1=Albert | last2=Laudenbach | first2=François | last3=Poénaru | first3=Valentin | title=Thurston's work on surfaces Translated from the 1979 French original by Djun M. Kim and Dan Margalit | series=Mathematical Notes | volume=48 | publisher=Princeton University Press | year=2012 | pages=xvi+254 |ISBN=978-0-691-14735-2}}
*{{cite book | ref=harv | last=Ivanov | first=Nikolai | title=Subgroups of Teichmüller Modular Groups | publisher=American Math. Soc. | year=1992}}
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