Faculty Publications

Combinatorial Modulus and Type of Graphs

Document Type

Article

Keywords

Discrete conformal geometry, Extremal length, Conformal modulus, Type problem, Outer spheres

Journal/Book/Conference Title

Topology and its Applications

Volume

156

Issue

17

First Page

2747

Last Page

2761

Abstract

Let A be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A', formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a disk triangulation graph may be refined without changing its combinatorial type, provided the refinement is not too wild. We also explore the type problem in terms of disk growth, proving a parabolicity condition based on a superlinear growth rate, which we also prove optimal. We prove our results with no degree restrictions in both the EEL and VEL settings and examine type problems for more general complexes and dual graphs.

Department

Department of Mathematics

Original Publication Date

11-1-2009

DOI of published version

10.1016/j.topol.2009.02.013

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