Thurston boundary

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.
The Thurston boundary of the Teichmüller space of a closed surface of genus formula_1 is homeomorphic to a sphere of dimension formula_2. The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary.
Measured foliations on surfaces.
Let formula_3 be a closed surface. A "measured foliation" formula_4 on formula_3 is a foliation formula_6 on formula_3 which may admit isolated singularities, together with a "transverse measure" formula_8, i.e. a function which to each arc formula_9 transverse to the foliation formula_6 associates a positive real number formula_11. 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.
Let formula_12 be the space of isotopy classes of closed simple curves on formula_3. A measured foliation formula_4 can be used to define a function formula_15 as follows: if formula_16 is any curve let
where the supremum is taken over all collections of disjoint arcs formula_18 which are transverse to formula_6 (in particular formula_20 if formula_16 is a closed leaf of formula_6). Then if formula_23 the intersection number is defined by:
Two measured foliations are said to be "equivalent" if they define the same function on formula_12 (there is a topological criterion for this equivalence via "Whitehead moves"). The space formula_26 of "projective measured laminations" is the image of the set of measured laminations in the projective space formula_27 via the embedding formula_28. If the genus formula_1 of formula_3 is at least 2, the space formula_26 is homeomorphic to the formula_2-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 formula_3 be a closed surface. Recall that a point in the Teichmüller space is a pair formula_34 where formula_35 is an hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to formula_36) and formula_37 a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be realised as a space of functionals on the set formula_12 of isotopy classes of simple closed curves on formula_12 as follows. If formula_40 and formula_23 then formula_42 is defined to be the length of the unique closed geodesic on formula_35 in the isotopy class formula_44. The map formula_45 is an embedding of formula_46 into formula_47, 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 formula_27 is still an embedding: let formula_49 denote the image of formula_46 there. Since this space is compact, the closure formula_51 is compact: it is called the "Thurston compactification" of the Teichmüller space.
The Thurston boundary.
The boundary formula_52 is equal to the subset formula_26 of formula_27. The proof also implies that the Thurston compactfification is homeomorphic to the formula_55-dimensional closed ball.
Applications.
Pseudo-Anosov diffeomorphisms.
A diffeomorphism formula_56 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 formula_57 respectively for some formula_58 (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 formula_59 is pseudo-Anosov if and only if:
The proof relies on the Brouwer fixed point theorem applied to the action of formula_59 on the Thurston compactification formula_51. 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 formula_3 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.
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 formula_68 can alternatively be seen as a faithful representation of the fundamental group formula_69 into the isometry group formula_70 of the hyperbolic plane formula_71, up to conjugation. Such an isometric action gives rise (via the choice of a principal ultrafilter) to an action on the asymptotic cone of formula_71, 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.