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.
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