Spheres in the curve complex

TL;DR

This paper investigates the geometry of metric spheres in the curve complex of a surface, revealing that generically, pairs of points on a sphere are almost always maximally distant, with distance twice the radius.

Contribution

It introduces a novel approach to analyze the geometry of the curve complex by defining null and generic sets, overcoming the lack of invariant measures.

Findings

Pairs of points on a sphere of radius R are almost always at distance 2R.

Metric spheres in the curve complex are countably infinite and lack invariant measures.

Most pairs of points on a sphere are maximally distant, indicating a specific geometric structure.

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.