Combinatorial modulus and type of graphs

Details Combinatorial modulus and type of graphs Combinatorial modulus and type of graphs

2006-08-25

arxiv.org/abs/math/0608624v2

Topology and its Applications Volume 156, Issue 17, 1 November 2009, Pages 2747-2761

Mathematics

Differential Geometry

24 pages, 12 figures

Scientific paper

10.1016/j.topol.2009.02.013

Let a $A$ 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.

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