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).
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
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
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.
· · ·
βr−1 β
· · ·
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
dSα (β, γ1) M + 4K.
For each β Sr(α), let E(β) = : (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
is null.
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