the case in the hyperbolic space
of any dimension endowed with the natural
measure on spheres. By contrast,
2 and
2 for all n and S,
with nontrivial dependence on S. In particular this shows that E(X) varies under
quasi-isometry, so it is a fine and not a coarse asymptotic statistic. Note that δ-
hyperbolicity itself (without an assumption of homogeneity) does not imply E = 2:
even for locally finite trees, one can get any value 0 E 2.
In [1], we show that for Teichm¨ uller space with the Teichm¨ uller metric and var-
ious visual measures on spheres, E(T (S)) = 2. Here, we show something stronger
for the curve graph.
Theorem. With respect to our notion of genericity for spheres in the curve
graph, almost every pair of points on Sr(α) is at distance exactly 2r apart.
This holds for every r and is certainly stronger than saying that the average
distance is asymptotic to 2r, so we can write E(C(S)) = 2. That is, suppose we start
with α and a pair of curves β and γ such that the shortest path in the curve graph
from α to either β or γ has length r. Then, almost surely, there is no more efficient
way to connect them with each other than to travel through α. This result tells us
that, even though the space C(S) is far from uniquely geodesic, the concatenation of
two geodesic segments of length r that share a common endpoint is almost always
itself geodesic. In this sense the curve graph is “even more hyperbolic than a tree”.
Of course, the meaningfulness of this result depends on the notion of genericity.
Lustig and Moriah [4] have introduced a very natural definition for genericity in
C(S) which uses the topology and measure class of PML(S). We identify the sphere
of radius 1 in C(S) with a lower-complexity curve complex, so that genericity can be
defined in the same way. We then extend to spheres of larger radius in a “visual”
manner; see Definition 3.2. While the Lustig–Moriah definition gives content to
statements about typical curves on S, our notion of genericity on spheres enables
us to talk about typical properties of high-distance curves on S.
2. Background
We fix a topological surface S = Sg,n with genus g and n punctures, and let
h = 6g 6 + 2n. Let S be the set of homotopy classes of essential nonperipheral
simple closed curves on S. From now on, a curve will mean an element of S. Next
we define the curve graph C(S): The vertex set of C(S) is S. In the case that h 2,
two curves are joined by an edge if they are disjointly realizable. In the case of S1,1
we join two vertices if the curves intersect once, and in the case of S0,4 two vertices
are joined by an edge if the curves intersect twice. In each of these cases, C(S) is
endowed with the standard path metric, denoted dS(α, β).
For α S, we write to denote the lower-complexity punctured (possibly
disconnected) surface obtained by cutting open S along α. Note that C(Sα) can be
realized as the subgraph of C(S) consisting of neighbors of α—that is, it is identified
with the sphere S1(α) C(S).
Recall that a measured lamination on S, given a hyperbolic structure, is a
foliation of a closed subset of S by geodesics, together with a measure on transver-
sals that is invariant under holonomy along the leaves of the lamination. We will
use ML(S) to denote the space of measured laminations on S. Let Mod(S) :=
denote the mapping class group of S; it acts on C(S) and on ML(S).
The latter has a natural Mod(S)-invariant measure μ. (This is the Lebesgue mea-
sure associated to the piecewise linear structure induced on ML(S) by train track
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