SOME DUALITY CONJECTURES FOR FINITE GRAPHS AND THEIR GROUP THEORETIC CONSEQUENCES

Abstract

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.

Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.

Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.