SPHERES IN THE CURVE COMPLEX 5

can be identified with a copy of PML(Sα); since this has lower dimension,

it has measure zero.

Because nullness is not preserved by countable unions, the following proposition

is less obvious. It is proved by Lustig–Moriah [4, Cor 5.3] using techniques from

handlebodies, but we include a proof for completeness. Our proof is similar to

well-known arguments, due to Luo and to Kobayashi [3], showing that the complex

of curves has infinite diameter (see the remarks following Proposition 4.6 of [5]).

Proposition 3.3. Any bounded-diameter subset of the curve graph C(S) is a

null subset of S.

We first prove that chains of disjoint laminations between two curves can be

realized with curves.

Definition 3.4. Given a pair of laminations F, G ∈ PML, define their inter-

section distance di(F, G) to be the smallest n such that there exist F = G0,G1,...,Gn =

G with i(Gj,Gj+1) = 0.

(We note that di(F, G) can be infinite; this occurs if at least one of them is

filling and they are topologically distinct.)

Lemma 3.5. di(α, β) = dS(α, β) for all α, β ∈ S.

Proof. Since simple closed curves are laminations, it is immediate that di(α, β) ≤

dS(α, β).

For the other direction, we can assume that the laminations are not filling.

Then given a lamination G ∈ PML, let us write YG for its supporting subsurface.

Now i(F, G) = 0 =⇒ i(F, ∂YG) = 0 =⇒ i(∂YF , ∂YG) = 0. But then given a

minimal-length disjointness path

α − G1 − G2 − · · · − Gn−1 − β,

we can realize it by simple closed curves by replacing each Gi

with ∂YGi .

Proof of Proposition 3.3. First given α ∈ S, let Sr(α) ⊂ C denote the

sphere of radius r centered at α. It is enough to prove the Proposition for each

Sr(α). For the ball of radius 1, the statement follows since each β satisfies i(α, β) =

0 and, as we saw above, the set of such β has measure 0 closure in PML(S). Notice

that this closure S1(α), which we identify with a copy of PML(Sα), is exactly the

set of laminations F ∈ PML(S) for which i(α, F ) = 0.

Now we consider the closure of the sphere of radius r and suppose G ∈ Sr(α).

Then G = limm→∞ βm r where βm r ∈ Sr(α). Then for each m there is a path

α, βm,...,βm

1 r

in C,where βm

j

∈ Sj(α). Passing to subsequences we can assume

that for each j the sequence βm j converges to some Gj ∈ Sj(α) with Gr = G.

Furthermore since i(βm,βm j j+1) = 0 it follows that

i(Gj,Gj+1) = 0.

Replacing each Gj with ∂YGj , as in the proof of Lemma 3.5, we see that i(G, γ) = 0

for some γ ∈ Sr−1(α). But then G ∈ S1(γ). Thus

Sr(α) ⊂

γ∈Sr−1(α)

S1(γ).

Thus Sr(α) is a countable union of measure-zero subsets of PML(S), hence has

zero measure itself. Therefore Sr(α) is a null set by definition.