Spotted disk and sphere graphs
Ursula Hamenst\"adt

TL;DR
This paper investigates the geometric properties of disk and sphere graphs associated with handlebodies, revealing the existence of high-dimensional Euclidean quasi-isometric embeddings, which inform the understanding of their large-scale geometry.
Contribution
It demonstrates that the disk graph for certain handlebodies contains quasi-isometric embeddings of and that the sphere graph of doubled handlebodies contains embeddings of ^n, highlighting their complex geometric structure.
Findings
Disk graph contains quasi-isometric embeddings for m=2.
Sphere graph contains quasi-isometric ^n embeddings for genus .
Reveals high-dimensional Euclidean structures within these graphs.
Abstract
The disk graph of a handlebody H of gneus with marked points on the boundary is the graph whose vertices are isotopy classes of disks disjoint from the marked points and where two vertices are connected by an edge of length one if they can be realized disjointly. We show that for m=2 the disk graph contains quasi-isometrically embedded copies of . Furthermore, the sphere graph of the doubled handlebody of genus with two marked points contains for every a quasi-isometrically embedded copy of .
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Taxonomy
TopicsGeometric and Algebraic Topology · Computational Geometry and Mesh Generation · Geometric Analysis and Curvature Flows
