2011/09/28 by Spencer Dowdall, Moon Duchin, Dowdall, Spencer +3
Computer Science · #57M50 #Data Management and Algorithms #Digital Image Processing Techniques #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1109.6338
openalex publication_date 2011/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.