Contemporary Mathematics
Volume 590, 2013
http://dx.doi.org/10.1090/conm/590/11727
Spheres in the curve complex
Spencer Dowdall, Moon Duchin, and Howard Masur
Abstract. In this paper we study the geometry of metric spheres in the
curve complex of a surface, with the goal of determining the “average” distance
between points on a given sphere. Averaging is not technically possible because
metric spheres in the curve complex are countably infinite and do not support
any invariant probability measures. To make sense of the idea of averaging,
we instead develop definitions of null and generic subsets in a way that is
compatible with the topological structure of the curve complex. With respect
to this notion of genericity, we show that pairs of points on a sphere of radius
r almost always have distance exactly 2r apart, which is as large as possible.
1. Introduction
The curve graph (or curve complex) C(S) associated to a surface S of finite
type is a locally infinite combinatorial object that encodes topological information
about the surface through intersection patterns of simple closed curves. It is known
to be δ-hyperbolic [5], a property that is often described by saying that a space is
“coarsely a tree”. To be precise, there exists δ such that for any geodesic triangle,
each side is in the δ-neighborhood of the union of the other two sides. In this note,
we will investigate the finer metric properties of the curve graph by considering the
geometry of spheres; specifically, we will study the average distance between pairs
of points on Sr(α), the sphere of radius r centered at α. To make sense of the
idea of averaging, we will develop a definition of null and generic sets in §3 that is
compatible with the topological structure of the curve graph.
Given a family of probability measures μr on the spheres Sr(x) in a metric space
(X, d), let E(X) = E(X, x, d, {μr}) be the normalized average distance between
points on large spheres:
E(X) := lim
r→∞
1
r
Sr(x)×Sr(x)
d(y, z) dμr(y)dμr(z),
if the limit exists. For finitely generated groups with their Cayley graphs, or more
generally for locally finite graphs, we can study averages with respect to counting
measure because the spheres are finite sets. It is shown in [2] that non-elementary
hyperbolic groups all have E(G, S) = 2 for any finite generating set S; this is also
2010 Mathematics Subject Classification. Primary 57M50,51F99.
The first author was partially supported by an NSF postdoctoral fellowship, MSPRF-1204814.
The second author was partially supported by NSF DMS-0906086.
The third author was partially supported by NSF DMS-0905907.
c 2013 American Mathematical Society
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