SPHERES IN THE CURVE COMPLEX 7

This suggests that the visual definition of nullness is better adapted to capturing

the geometry of spheres in certain spaces, while the inductive definition would be

better adapted to others. However, being null in our sense implies nullness in the

weaker sense.

Returning to the curve graph: consider distinct curves β, γ ∈ S1(α). Clearly

dS(β, γ) is either 1 or 2. We can easily see that such pairs generically have distance

2 because since α and β are disjoint, the set of γ ∈ S1(α) for which dS(β, γ) = 1 is

contained in PML(Sα,β), which has codimension at least two in PML(Sα).

β

γ

α

Our main result is that given any limit on their length, paths connecting points

on the sphere “almost surely” pass through the center.

Theorem 3.9 (Avoiding the center). For a surface S with h ≥ 4, consider a

point α ∈ C(S). For K 0, let Pr

K(α)

⊆ Sr(α)×Sr(α) consist of those pairs (β, γ)

that are connected by some path of length ≤ K that does not go through α. Then

for any K and r, the set Pr

K(α) is null.

Proof. For any pair (β, γ) ∈ Pr

K(α),

there is a path β = δ0,δ1,...,δk = γ

in C(S) with k ≤ K, and δi = α for each i. Two successive curves δi and δi+1,

since they are disjoint and intersect Sα, project to nonempty sets in C(Sα) whose

distance from each other is at most 4; thus dSα (β, γ) ≤ 4K.

α

β1

· · ·

βr−1 β

γ1

· · ·

γr−1 γ

Let β1,γ1 be any closest points on S1(α) to β, γ, respectively. Since we can

join γ, γ1 by a geodesic in C(S) that misses α, there is a constant M = M(S)

coming from Masur-Minsky [6, Thm 3.1] such that dSα (γ, γ1) ≤ M. By the triangle

inequality,

dSα (β, γ1) ≤ M + 4K.

For each β ∈ Sr(α), let E(β) = {γ : (β, γ) ∈ Pr K(α)} and then consider the

corresponding E1(β) = {γ1 ∈ S1(α) : E(β) ∩ Sr−1(γ1) = ∅}.

We have shown that E1(β) has diameter at most 2M + 8K by the triangle

inequality and is therefore null in S1(α) by Proposition 3.3. Thus E(β) is null for

all β, so Pr

K(α)

is null.