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 g {\displaystyle g} is homeomorphic to a sphere of dimension 6 g − 7 {\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 S {\displaystyle S} be a closed surface. A measured foliation ( F , μ ) {\displaystyle ({\mathcal {F}},\mu )} on S {\displaystyle S} is a foliation F {\displaystyle {\mathcal {F}}} on S {\displaystyle S} which may admit isolated singularities, together with a transverse measure μ {\displaystyle \mu } , i.e. a function which to each arc α {\displaystyle \alpha } transverse to the foliation F {\displaystyle {\mathcal {F}}} associates a positive real number μ ( α ) {\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. Let S {\displaystyle {\mathcal {S}}} be the space of isotopy classes of closed simple curves on S {\displaystyle S} . A measured foliation ( F , μ ) {\displaystyle ({\mathcal {F}},\mu )} can be used to define a function i ( ( F , μ ) , ⋅ ) ∈ R + S {\displaystyle i(({\mathcal {F}},\mu ),\cdot )\in \mathbb {R} _{+}^{\mathcal {S}}} as follows: if γ {\displaystyle \gamma } is any curve let
μ ( γ ) = sup α 1 , … , α r ( ∑ i = 1 r μ ( α i ) ) {\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 α 1 … , α r ⊂ γ {\displaystyle \alpha _{1}\ldots ,\alpha _{r}\subset \gamma } which are transverse to F {\displaystyle {\mathcal {F}}} (in particular μ ( γ ) = 0 {\displaystyle \mu (\gamma )=0} if γ {\displaystyle \gamma } is a closed leaf of F {\displaystyle {\mathcal {F}}} ). Then if σ ∈ S {\displaystyle \sigma \in {\mathcal {S}}} the intersection number is defined by:
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