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.
The Thurston boundary of the Teichmüller space of a closed surface of genus
The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary.
which may admit isolated singularities, together with a transverse measure
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.
be the space of isotopy classes of closed simple curves on
is any curve let where the supremum is taken over all collections of disjoint arcs
(there is a topological criterion for this equivalence via Whitehead moves).
of projective measured foliations is the image of the set of measured foliations in the projective space
Recall that a point in the Teichmüller space is a pair
is a hyperbolic surface (a Riemannian manifold with sectional curvatures all equal to
a homeomorphism, up to a natural equivalence relation.
of isotopy classes of simple closed curves on
is defined to be the length of the unique closed geodesic on
is compact: it is called the Thurston compactification of the Teichmüller space.
The proof also implies that the Thurston compactification is homeomorphic to the
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
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.
is pseudo-Anosov if and only if: The proof relies on the Brouwer fixed point theorem applied to the action of
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.
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] The compactification of Teichmüller space by adding measured foliations is essential in the definition of the ending laminations of a hyperbolic 3-manifold.
can alternatively be seen as a faithful representation of the fundamental group
Such an isometric action gives rise (via the choice of a principal ultrafilter) to an action on the asymptotic cone of
Two such actions are equivariantly isometric if and only if they come from the same point in Teichmüller space.