SPHERES IN THE CURVE COMPLEX 3
neighborhoods.) The space of projective measured laminations PML(S) is obtained
by identifying laminations whose transverse measures differ only by a scalar multi-
ple; it is endowed with the topology of a sphere of dimension h − 1 and it inherits
a natural Mod(S)-invariant measure class, which we will again denote by μ.
A train track on S is a finite collection of disjointly embedded arcs, called
branches, meeting at vertices, called switches, such that the branches are C1 away
from switches and have well-defined tangents at the switches. (There are also non-
degeneracy conditions for switches and topological conditions on the complement;
see  for details.) Train tracks exist on every surface with h 0, and we say that
a lamination F is carried by a train track τ if there is a map φ : S → S isotopic to
the identity with φ(F ) ⊂ τ . Via this carrying relation, measured laminations that
are carried by τ correspond to choices of weights on the branches of τ that satisfy
A connected proper subsurface V of S is essential if all components of ∂V are
essential; i.e., they are homotopically nontrivial and not isotopic to a puncture.
Definition 2.1. Consider a non-annular essential subsurface V . The subsur-
face projection πV is a coarsely well-defined map C(S) → C(V ) defined as follows.
Realize β ∈ S and ∂V as geodesics (in any hyperbolic metric on S). If β ⊂ V , let
πV (β) = β. If β is disjoint from V , then πV (β) is undefined. Otherwise, β ∩ V is
a disjoint union of finitely many homotopy classes of arcs with endpoints on ∂V ,
and we obtain πV (β) by choosing any arc and performing a surgery along ∂V to
create a simple closed curve contained in V . All possible ways to do this form a
non-empty subset of the curve complex C(V ) with uniformly bounded diameter.
We can denote by dV (α, β) the diameter in C(V ) of πV (α) ∪ πV (β). If α and β are
disjoint and both intersect V then dV (α, β) ≤ 4.
There is a well-defined inclusion S → PML(S) whose image is dense and we
will identify S with its image under that map. The supporting subsurface of a
lamination is the subsurface filled by F . We will denote the geometric intersection
number on S by i(α, β), and we recall that it has a well-defined extension to ML(S).
On PML(S), we can thus talk about whether or not i(F, G) = 0. Then PML(Sα)
can be identified with the subset of PML(S) consisting of those laminations F for
which i(F, α) = 0. If α is nonseparating, then Sα has complexity h − 2; if α is
separating, then Sα consists of two surfaces with complexity h1 + h2 = h − 2. In
that case we consider PML(Sα) as a product of the corresponding spaces for the
two components. In either case we see that PML(Sα) has positive codimension in
Definition 2.2. Given a group G that acts on a space X with Borel algebra
B(X), a G-invariant mean on X is a function σ : B(X) → [0, 1] such that
• σ(∅) = 0 and σ(X) = 1 ;
• if B1,...BN ∈ B(X) are pairwise disjoint then σ( Bi) =
• σ(B) = σ(gB) for all B ∈ B(X) and all g ∈ G.
Note that invariant means are only required to be finitely additive, while mea-
sures must be countably additive.
Proposition 2.3. There is no Mod(S)-invariant mean on S or on PML.