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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